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arXiv:astro-ph/0506751v1 30 Jun 2005
Astronomy & Astrophysics
manuscript no. ga˙ostars February 5, 2008
(DOI: will be inserted by hand later)
Spectral analysis of early-type stars using a
genetic algorithm based fitting method
M. R. Mokiem1, A. de Koter1, J. Puls2, A. Herrero3,4, F. Najarro5, and M. R. Villamariz3
1Astronomical Institute Anton Pannekoek, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands
2Universit¨ats-Sternwarte M¨unchen, Scheinerstr. 1, D-81679 M¨unchen, Germany
3Instituto de Astrof´ısica de Canarias, E-38200, La Laguna, Tenerife, Spain
4Departamento de Astrof´ısica, Universidad de La Laguna, Avda. Astrof´ısico Francisco S´anchez, s/n, E-38071 La Laguna,
Spain
5Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cient´ıficas, CSIC, Serrano 121, E-28006 Madrid,
Spain
Received /Accepted
Abstract. We present the first automated fitting method for the quantitative spectroscopy of O- and early B-type stars with
stellar winds. The method combines the non-LTE stellar atmosphere code from Puls et al. (2005) with the genetic
algorithm based optimizing routine from Charbonneau (1995), allowing for a homogeneous analysis of upcoming large
samples of early-type stars (e.g. Evans et al. 2005). In this first implementation we use continuum normalized optical hydrogen
and helium lines to determine photospheric and wind parameters. We have assigned weights to these lines accounting for line
blends with species not taken into account, lacking physics, and/or possible or potential problems in the model atmosphere
code. We find the method to be robust, fast, and accurate. Using our method we analysed seven O-type stars in the young
cluster Cyg OB2 and five other Galactic stars with high rotational velocities and/or low mass loss rates (including 10 Lac,
ζOph, and τSco) that have been studied in detail with a previous version of . The fits are found to have a quality that
is comparable or even better than produced by the classical “by eye” method. We define errorbars on the model parameters
based on the maximum variations of these parameters in the models that cluster around the global optimum. Using this concept,
for the investigated dataset we are able to recover mass-loss rates down to ∼6×10−8M⊙yr−1to within an error of a factor
of two, ignoring possible systematic errors due to uncertainties in the continuum normalization. Comparison of our derived
spectroscopic masses with those derived from stellar evolutionary models are in very good agreement, i.e. based on the limited
sample that we have studied we do not find indications for a mass discrepancy. For three stars we find significantly higher
surface gravities than previously reported. We identify this to be due to differences in the weighting of Balmer line wings
between our automated method and “by eye” fitting and/or an improved multidimensional optimization of the parameters. The
empirical modified wind momentum relation constructed on the basis of the stars analysed here agrees to within the error bars
with the theoretical relation predicted by Vinket al. (2000), including those cases for which the winds are weak (i.e. less than
a few times 10−7M⊙yr−1).
Key words. methods: data analysis - line: profiles - stars: atmospheres - stars: early-type - stars: fundamental parameters -
stars: mass loss
1. Introduction
Until about a decade ago detailed analysis of the photospheric
and wind properties of O-type stars was limited to about 40 to
50 stars divided over the Galaxy and the Magellanic Clouds
(see e.g. Puls et al. 1996; see also Repolust, Puls, & Herrero
2004). The reason that at that time only such a limited number
of objects had been investigated is related in part to the fact that
considerable effort was directed towards improving the physics
of the non-local thermodynamic equilibrium (non-LTE) model
atmospheres used to analyse massive stars. Notable devel-
Send offprint requests to: M.R. Mokiem,
e-mail: mokiem@science.uva.nl
opments have been the improvements in the atomic models
(e.g. Becker & Butler 1992), shock treatment (Pauldrach et al.
2001), clumping (Hillier 1991; Hillier & Miller 1999), and the
implementation of line blanketing (e.g. Hubeny & Lanz 1995;
Hillier & Miller 1998; Pauldrach et al. 2001). To study the ef-
fects of these new physics a core sample of “standard” O-type
stars has been repeatedly re-analysed. A second reason, that
is at least as important, is the complex, and time and CPU
intensive nature of these quantitative spectroscopic analyses.
Typically, at least a six dimensional parameter space has to
be probed, i.e. effective temperature, surface gravity, helium
to hydrogen ratio, atmospheric microturbulent velocity, mass-
loss rate, and a measure of the acceleration of the transonic
2 M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
outflow. Rotational velocities and terminal outflow velocities
can be determined to considerable accuracy by means of ex-
ternal methods such as rotational (de-) convolution methods
(e.g. Howarth et al. 1997) and SEI-fitting of P-Cygni lines (e.g.
Groenewegen & Lamers 1989), respectively. To get a good
spectral fit it typically requires tens, sometimes hundreds of
models per individual star.
In the last few years the field of massive stars has seen
the fortunate development that the number of O-type stars that
have been studied spectroscopically has been doubled (e.g.
Crowther et al. 2002; Herrero et al. 2002; Bianchi & Garcia
2002; Bouret et al. 2003; Hillier et al. 2003; Garcia & Bianchi
2004; Martins et al. 2004; Massey et al. 2004; Evans et al.
2004). The available data set of massive O- and early B-type
stars has recently again been doubled, mainly through the ad-
vent of multi-object spectroscopy. Here we explicitly men-
tion the VLT-FLAMES Survey of Massive Stars (Evans et al.
2005) comprising over 100 hours of VLT time. In this survey
multi-object spectroscopy using the Fibre Large Array Multi-
Element Spectrograph (FLAMES) has been used to secure over
550 spectra (of which in excess of 50 are spectral type O) in
a total of seven clusters distributed over the Galaxy and the
Magellanic Clouds.
This brings within reach different types of studies that so
far could only be attempted with a troublingly small sample
of stars. These studies include establishing the mass loss be-
haviour of Galactic stars across the upper Hertzsprung-Russell
diagram, from the weak winds of the late O-type dwarfs (of
order 10−8M⊙yr−1) to the very strong winds of early O-type
supergiants (of order 10−5M⊙yr−1); determination of the mass-
loss versus metallicity dependence in the abundance range
spanned by Small Magellanic Cloud to Galactic stars; placing
constraints on the theory of massive star evolution by compar-
ing spectroscopic mass determinations and abundance patterns
with those predicted by stellar evolution computations, and the
study of (projected) spatial gradients in the mass function of
O- and B-type stars in young clusters, as well as such spatial
gradients in the initial atmospheric composition of these stars.
To best perform studies such as listed above not only re-
quires a large set of young massive stars, it also calls for
a robust, homogeneous and objective means to analyse such
datasets using models that include state-of-the-art physics. This
essentially requires an automated fitting method. Such an auto-
mated method should not only be fast, it must also be suffi-
ciently flexible to be able to treat early-type stars with widely
different properties (e.g. mass-loss rates that differ by a factor
of 103). Moreover, it should apply a well defined fitting cri-
terium, like a χ2criterium, allowing it to work in an automated
and reproducible way.
To cope with the dataset provided by the VLT-FLAMES
Survey and to improve the objectivity of the analysis, we
have investigated the possibility of automated fitting. Here we
present a robust, fast, and accurate method to perform auto-
mated fitting of the continuum normalized spectra of O- and
early B-type stars with stellar winds using the fast performance
stellar atmosphere code (Puls et al. 2005) combined
with a genetic algorithm based fitting method. This first imple-
mentation of an automated method should therefore be seen as
an improvement over the standard “by eye” method, and not as
a replacement of this method. The improvement lies in the fact
that with the automated method large data sets (tens or more
stars), spanning a wide parameter space, can be analysed in a
repeatable and homogeneous way. It does not replace the “by
eye” method as our automated fitting method still requires a
by eye continuum normalization as well as a human controlled
line selection. This latter should address the identification and
exclusion of lines that are not modeled (i.e. blends), as well as
introduce information on lacking physics and/or possible or po-
tential problems in the model atmosphere code. Future imple-
mentations of an automated fitting method may use the abso-
lute spectrum, preferably over a broad wavelength range. This
would eliminate the continuum rectification problem, however,
it will require a modeling of the interstellar extinction. In this
way one can work towards a true replacing of the “by eye”
method by an automated approach.
In Sect. 2 we describe the genetic algorithm method and
implementation, and we provide a short r´esum´e of the applied
unified, non-LTE, line-blanketed atmosphere code –
which is the only code to date for which the method described
here is actually achievable (in the context of analysing large
data sets). To test the method we analyse a set of 12 early type
spectra in Sect. 3. We start with a re-analysis of a set of seven
stars in the open cluster Cyg OB2 that have been studied by
Herrero et al. (2002). The advantage of focusing on this cluster
is that it has been analysed with a previous version of ,
allowing for as meaningful a comparison as is possible, while
still satisfying our preference to present a state-of-the-art anal-
ysis. The analysis of Cyg OB2 has the added advantage that all
stars studied are approximately equidistant. To test the perfor-
mance of our method outside the parameter range offered by
the Cyg OB2 sample we have included an additional five well-
studied stars with either low density winds and/or very high
rotational velocities. In Sect. 4 we describe our error analy-
sis method for the multidimensional spectral fits obtained with
the automated method. A systematic comparison of the ob-
tained parameters with previously determined values is given
in Sect. 5. Implications of the newly obtained parameters on
the properties of massive stars are discussed in Sect. 6. In the
last section we give our conclusions.
2. Automated fitting using a genetic algorithm
2.1. Spectral line fitting as an optimization problem
Spectral line fitting of early-type stars is an optimization prob-
lem in the sense that one tries to maximize the correspondence
between a given observed spectrum and a synthetic spectrum
produced by a stellar atmosphere model. Formally speaking
one searches for the global optimum, i.e. best fit, in the param-
eter space spanned by the free parameters of the stellar atmo-
sphere model by minimizing the differences between the ob-
served and synthesized line profiles.
Until now the preferred method to achieve this minimiza-
tion has been the so called fitting “by eye” method. In this
method the best fit to the observed spectrum of a certain ob-
ject is determined in an iterative manner. Starting with a first
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 3
guess for the model parameters a spectrum is synthesized. The
quality of the fit to the observed spectrum is determined, as
is obvious from the methods name, by an inspection by eye.
Based on what the person performing the fit sees, for instance,
whether the width of the line profiles are reproduced correctly,
combined with his/her experience and knowledge of the model
and the object, the model parameters are modified and a new
spectrum is synthesized. This procedure is repeated until the
quality of the fit determined by eye cannot be increased any-
more by modifying the model parameters.
It can be questioned whether a fit constructed with the fit-
ting “by eye” method corresponds to the best fit possible, i.e.
the global optimum. Reasons for this are, i) the restricted size
of parameter space that can be investigated, both in terms of
number of free parameters as well as absolute size of the pa-
rameter domain that can investigated with high accuracy, ii) the
limited number of free parameters that are changed simultane-
ously, and iii) biases introduced by judging the quality of a line
fit by eye. The importance of the first point lies in the fact that
in order to assure that the global optimum is found, a param-
eter space that is as large as possible should be explored with
the same accuracy for all parameters in the complete parame-
ter space. If this is not the case the solution found will likely
correspond to a local optimum.
The argument above becomes stronger in view of the sec-
ond point. Spectral fitting is a multidimensional problem in
which the line profile shapes depend on all free parameters si-
multaneously, though to a different extent. Consequently, the
global optimum can only be found if all parameters are al-
lowed to vary at the same time. The use of fit diagrams (e.g.
Kudritzki & Simon 1978; Herrero et al. 1992) does not resolve
this issue. These diagrams usually only take variations in Teff
and log ginto account, neglecting the effects of other pa-
rameters, like microturbulence (e.g. Smith & Howarth 1998;
Villamariz & Herrero 2000) and mass loss (e.g. [Fig. 5 of]
Mokiem et al. 2004), on the line profiles.
The last point implies that, strictly speaking, fitting “by
eye” cannot work in a reproducible way. There is no uniform
well defined method to judge how well a synthetic line pro-
file fits the data by eye. More importantly, it implies that there
is no guarantee that the synthetic line profiles selected by the
eye, correspond to the profiles which match the data the best.
This predominantly increases the uncertainty in the derivation
of those parameters that very sensitively react to the line profile
shape, like for instance the surface gravity.
The new fitting method presented here does not suffer from
the drawbacks discussed above. It is an automated method ca-
pable of global optimization in a multi-dimensional parameter
space of arbitrary size (Sect. 2.5). As it is automated, it does not
require any human intervention in finding the best fit, avoiding
potential biases introduced by “by eye” interpretations of line
profiles. The method described here consists of two main com-
ponents. The first component is the non-LTE stellar atmosphere
code . Section 2.4 gives an overview of the capabilities
of the code and the assumptions involved. The second compo-
nent is the genetic algorithm (GA) based optimizing routine
from Charbonneau (1995), which is responsible for op-
timizing the parameters of the models. For the tech-
nical details of this routine and more information on GAs we
refer to the cited paper and references therein. Here we will suf-
fice with a short description of GAs and a description of the GA
implementation with respect to optimization of spectral fits.
2.2. The genetic algorithm implementation
Genetic algorithms represent a class of heuristic optimization
techniques, which are inspired by the notion of evolution by
means of natural selection (Darwin 1859). They provide a
method of solving optimization problems by incorporating this
biological notion in a numerical fashion. This is achieved by
evolving the global solution over subsequent generations start-
ing from a set of randomly guessed initial solutions, so called
individuals. Selection pressure is imposed in between genera-
tions based on the quality of the solutions, their so called fit-
ness. A higher fitness implies a higher probability the solution
will be selected for reproduction. Consequently, only a selected
set of individuals will pass on their “genetic material” to sub-
sequent new generations.
To create the new generations discussed above GAs require
a reproduction mechanism. In its most basic form this mecha-
nism consists of two genetic operators. These are the crossover
operator, simulating sexual reproduction, and the mutation op-
erator, simulating copying errors and random effects affecting
a gene in isolation. An important benefit of these two opera-
tors is the fact that they also introduce new genetic material
into the population. This allows the GA to explore new re-
gions of parameters space, which is important in view of the
existence of local extremes. When the optimization runs into
a local optimum, these two operators, where usually muta-
tion has the strongest effect, allow for the construction of in-
dividuals outside of this optimum, thereby allowing it to find
a path out of the local optimum. This capability to escape lo-
cal extremes, consequently, classifies GAs as global optimiz-
ers and is one of the reasons they have been applied to many
problems in and outside astrophysics (e.g. Metcalfe et al. 2000;
Gibson & Charbonneau 1998).
Using an example we can further illustrate the GA opti-
mization technique. Lets assume that the optimization problem
is the minimization of some function f. This function has n
variables, serving as the genetic building blocks, spanning a
ndimensional parameter space. The first step in solving this
problem is to create an initial population of individuals, which
are sets of nparameters, randomly distributed in parameter
space. For each of these individuals the quality of their solution
is determined by simply calculating ffor the specific param-
eter values. Now selection pressure is imposed and the fittest
individuals, i.e. those that correspond to the lowest values of f,
are selected to construct a new generation. As the selected in-
dividuals represent the fittest individuals from the population,
every new generation will consist of fitter individuals, leading
to a minimization of f, thereby, solving the optimization prob-
lem.
With the previous example in mind we can explain our im-
plementation of the GA for solving the optimization problem
of spectral line fitting, with the following scheme. We start out
4 M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
with a first generation of a population of models ran-
domly distributed in the free parameter space (see Sect. 2.5).
For each of these models it is determined how well an observed
spectrum is fitted by calculating the reduced chi squared, χ2
red,i,
for each of the fitted lines i. The fitness F, of a model is then
defined as the inverted sum of the χ2
red,i’s, i.e.
F≡
N
X
i
χ2
red,i
−1
,(1)
where Ncorresponds to the number of lines evaluated. The
fittest models are selected and a new generation of models is
constructed based on their parameters. From this generation the
fitnesses of the models are determined and again from the fittest
individuals a new generation is constructed. This is repeated
until Fis maximized, i.e. a good fit is obtained.
In terms of quantifying the fit quality Eq. (1) does not rep-
resent a unique choice. Other expressions for the fitness cri-
terium, for instance, the sum of the inverted χ2
red,i’s of the indi-
vidual lines, or the inverted χ2
red of all the spectral points eval-
uated, also produce the required functionality of an increased
fitness with an increased fit quality. We have chosen this par-
ticular form based on two of its properties. Firstly, the eval-
uation of the fit quality of the lines enter into the expression
individually, ensuring that, regardless of the number of points
in a certain line, all lines are weighted equally. This allows as
well for weighting factors for individual lines, which express
the quality with which the stellar atmosphere synthesizes these
lines (cf. Sect. 3.2). Secondly, using the inverted sum of the
χ2
red,i’s instead of the sum of the inverted χ2
red,i’s avoids having
a single line, which is fitted particularly well, to dominate the
solution. Instead the former form demands a good fit of all lines
simultaneously.
2.3. Parallelization of the genetic algorithm
The ability of global optimization of GAs comes at a price.
Finding the global minimum requires the calculation of many
generations. In Sect. 2.6 we will show that for the spectra stud-
ied in this paper, the evaluation of more than a hundred genera-
tions is needed to assure that the global optimum is found. For
a typical population size of ∼70 individuals, this comes down
to the calculation of ∼7000 models. With a modern
3 GHz processor a single model (aiming at the analy-
sis of hydrogen and helium lines) can be calculated within five
to ten minutes. Consequently, automated fitting on a sequential
computer would be unworkable.
To overcome this problem, parallelization of the rou-
tine is necessary. This parallelization is inspired by the work
of Metcalfe & Charbonneau (2003). Consequently, our paral-
lel version is very similar to the version of these authors. The
main difference between the two versions, is an extra paral-
lelization of the so called elitism option in the reproduction
schemes (see Metcalfe & Charbonneau). This was treated in a
sequential manner in the Metcalfe & Charbonneau implemen-
tation and has now been parallelized as well.
Due to the strong inherent parallelism of GAs, the paral-
lel version of our automated fitting method scales very well
with the number of processors used. Test calculations showed
that for configurations in which the population size is an integer
multiple of the number of processors the sequential overhead is
negligible. Consequently, the runtime scales directly with the
inverse of the number of processors. Thus enabling the auto-
mated fitting of spectra.
2.4. The non-LTE model atmosphere code
For modeling the optical spectra of our stars we use the latest
version of the non-LTE, line-blanketed atmosphere code -
for early-type stars with winds. For a detailed description
we refer to Puls et al. (2005). Here we give a short overview of
the assumptions made in this method. The code has been devel-
oped with the emphasis on a fast performance (hence its name),
which makes it currently the best suited (and realistically only)
model for use in this kind of automated fitting methods.
adopts the concept of “unified model atmo-
spheres”, i.e. including both a pseudo-hydrostatic photosphere
and a transonic stellar wind, assuring a smooth transition be-
tween the two. The photospheric density structure follows from
a self-consistent solution of the equation of hydrostatic equilib-
rium and accounts for the actual temperature stratification and
radiation pressure. The temperature calculation utilizes a flux-
correction method in the lower atmosphere and the thermal bal-
ance of electrons in the outer atmosphere (with a lower cut-off
at Tmin =0.5Teff). In the photosphere the velocity structure,
v(r), corresponds to quasi-hydrostatic equilibrium; outside of
this regime, in the region of the sonic velocity and in the super-
sonic wind regime it is prescribed by a standard β-type velocity
law, i.e.
v(r)=v∞1−r◦
rβ
,(2)
where v∞is the terminal velocity of the wind. The parameter
r◦is used to assure a smooth connection, and βis a measure of
the flow acceleration.
The code distinguishes between explicit elements (in our
case hydrogen and helium) and background elements (most im-
portantly: C, N, O, Ne, Mg, Si, S, Ar, Fe, Ni). The explicit el-
ements are used as diagnostic tools and are treated with high
precision, i.e. by detailed atomic models and by means of co-
moving-frame transport for the line transitions. The H and
He model atoms consist of 20 levels each; the He model in-
cludes levels up to and including n=10, where levels with
n≥8 have been packed. The background ions are included to
allow for the effects of line-blocking (treated in an approximate
way by using suitable means for the corresponding line opaci-
ties) and line-blanketing. Occupation numbers and opacities of
both the explicit and the most abundant background ions are
constrained by assuming statistical equilibrium. The only dif-
ference between the treatment of these types of ions is that for
the background ions the Sobolev approximation is used in de-
scribing the line transfer (accounting for the actual illumination
radiation field).
Abundances of the background elements are taken from the
solar values provided by Grevesse & Sauval (1998, and refer-
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 5
ences therein). The He/H ratio is not fixed and can be scaled
independently from the background element abundances.
A comparison between the optical H and He lines as syn-
thesized by and those predicted by the independent
comparison code (Hillier & Miller 1998) show excel-
lent agreement, save for the He singlet lines in the temperature
range between 36 000 and 41 000 K for dwarfs and between
31 000 and 35 000 K for supergiants, where predicts
weaker lines. We give account of this discrepancy, and there-
fore of an increased uncertainty in the reproduction of these
lines, by introducing weighting factors, which for the He sin-
glets of stars in these ranges are lower (cf. Sect. 3.2).
2.5. Fit parameters
The main parameters which will be determined from a spectral
fit using are the effective temperature Teff, the surface
gravity g, the microturbulent velocity vturb, the helium over hy-
drogen number density YHe, the mass loss rate ˙
Mand the expo-
nent of the beta-type velocity law β. These parameters span the
free parameter space of our fitting method. The stellar radius,
R⋆, is not a free parameter as its value is constrained by the
absolute visual magnitude MV. To calculate R⋆we adopt the
procedure outlined in Kudritzki (1980), i.e.
5 log R/R⊙=29.57 −(MV−V),(3)
where Vis the visual flux of the theoretical model given by
−2.5 log Z∞
0FλSλdλ . (4)
In the above equation Sλis the V-filter function of
Matthews & Sandage (1963) and Fλis the theoretical stellar
flux. Note that as R⋆is an input parameter, Fλis not known
before the model is calculated. Therefore, during the
automated fitting we approximate Fλby a black body radiating
at T=0.9Teff(cf. Markova et al. 2004). After the fit is com-
pleted we use the theoretical flux from the best fit model to cal-
culate the non approximated stellar radius. Based on this radius
we rescale the mass loss rate using the invariant wind-strength
parameter Q(Puls et al. 1996; de Koter et al. 1997)
Q=˙
M
(v∞R⋆)3
2
.(5)
The largest difference between the approximated and final stel-
lar radius for the objects studied here, is ∼2 percent. The corre-
sponding rescaling in ˙
Mis approximately three percent.
The projected rotation velocity, vrsin i, and terminal veloc-
ity of the wind are not treated as free parameters. The value of
vrsin iis determined from the broadening of weak metal lines
and the width of the He lines. For v∞we adopt values ob-
tained from the study of ultraviolet (UV) resonance lines, or, if
not available, values from calibrations are used.
Our fitting method only requires the size of the free param-
eter domain to be specified. For the objects studied in this paper
we keep the boundaries between which the parameters are al-
lowed to vary, fixed for vturb,YHe and β. The adopted ranges,
respectively, are [0, 20] km s−1, [0.05, 0.30] and [0.5, 1.5]. The
boundaries for Teffare set based on the spectral type and lumi-
nosity class of the studied object. Usually the size of this range
is set to approximately 5000 K. The log grange is delimited
so that the implied stellar mass lies between reasonable bound-
aries. For instance for the B1 I star Cyg OB2 #2 the adopted Teff
range together with its absolute visual magnitude imply a pos-
sible range in R⋆of [11.5:12.0]R⊙. For the automated fit we set
the minimum and maximum log gto 3.1 and 3.8, respectively,
which sets the corresponding mass range that will be investi-
gated to [5.0:25.2] M⊙. For the mass loss rate we adopt a con-
servative range of at least one order of magnitude. As example
for the analysis of Cyg OB2 #2 we adopted lower and upper
boundaries of 4 ×10−8and 2 ×10−6M⊙yr−1, respectively.
2.6. Formal tests of convergence
Before we apply our automated fitting method to real spec-
tra, we first test whether the method is capable of global opti-
mization. For this we perform convergence tests using synthetic
data. The main goal of these tests is to determine how well and
how fast the input parameters, used to create the synthetic data,
can be recovered with the method. The speed with which the
input parameters are recovered, i.e. the number of generations
needed to find the global optimum, can then be used to deter-
mine how many generations are needed to obtain the best fit for
a real spectrum. In other words, when the fit has converged to
the global optimum.
Three synthetic datasets, denoted by A, B and C, were cre-
ated with the following procedure. First, line profiles of Balmer
hydrogen lines and helium lines in the optical blue and Hα
in the red calculated by were convolved with a ro-
tational broadening profile. Table 1 lists the parameters of the
three sets of models as well as the projected rotational veloc-
ity used. A second convolution with a Gaussian instrumental
profile was applied to obtain a spectral resolution of 0.8 Å and
1.3 Å for, respectively, the Hαline and all other lines. These
values correspond to the minimum resolution of the spectra fit-
ted in Sect. 3. Finally, Gaussian distributed noise, correspond-
ing to a signal to noise value of 100, was added to the profiles.
Dataset A represents an O3 I star with a very dense stellar wind
(˙
M=10−5M⊙yr−1), while set B is that of an O5.5 I with a
more typical O-star mass loss. The last set C is characteristic
for a B0 V star with a very tenuous wind of only 10−8M⊙yr−1.
From the synthetic datasets we fitted nine lines, three hy-
drogen, three neutral helium and three singly ionized helium
lines, corresponding to the minimum set of lines fitted for a
single object in Sect. 3. The fits were obtained by evolving a
population of 72 models over a course of 200 genera-
tions. In this test and throughout the remainder of the paper we
use with a dynamically adjustable mutation rate, with the
minimum and maximum mutation rate set to the default values
(see Charbonneau & Knapp 1995). Selection pressure, i.e. the
weighting of the probability an individual will be selected for
reproduction based on its fitness, was also set to the default
value.
Table 1 lists the parameter ranges in which the method was
allowed to search, i.e. the minimum and maximum values al-
6 M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
Table 1. Input parameters of the formal test models (“In” column) and parameters obtained with the automated fitting method by
fitting synthetic data created from these models (“Out” column). Results were obtained by evolving a population of 72
models over 200 generations.
Set A Search Set B Search Set C Search
In range Out In range Out In range Out
Spectral type O3 I O5.5 I B0 V
Teff[kK] 45.0 [42, 47] 45.0 37.5 [35, 40] 37.6 30.0 [28, 34] 29.9
log g[cm s−2] 3.80 [3.5, 4.0] 3.84 3.60 [3.3, 3.9] 3.57 4.00 [3.7, 4.3] 3.95
R⋆[R⊙] 17.0 20.0 8.0
log L⋆[L⊙] 6.03 - 5.85 - 4.67 -
vturb [km s−1] 5.0 [0, 20] 5.9 10.0 [0, 20] 9.7 15.0 [0, 20] 14.8
YHe 0.15 [0.05, 0.30] 0.15 0.10 [0.05, 0.30] 0.10 0.10 [0.05, 0.30] 0.10
˙
M[10−6M⊙yr−1] 10.0 [1.0, 20.0] 9.3 5.0 [1.0, 10.0] 5.3 0.01 [0.001, 0.2] 0.008
β1.20 [0.5, 1.5] 1.18 1.00 [0.5, 1.5] 0.99 0.80 [0.5, 1.5] 0.93
v∞[km s−1] 2500 - 2200 - 2000 -
vrsin i[km s−1] 150 - 120 - 90 -
lowed for the parameters of the models. As v∞and
vrsin iare not free parameters these were set equal to the input
values.
In all the three test cases the automated method was able to
recover the global optimum. Table 1 lists the parameters of the
best fit models obtained by the method in the “Out” columns.
Compared to the parameters used to create the synthetic data,
there is very good agreement. Moderate differences (of a 15-
20% level) are found for vturb recovered from dataset A and
for the wind parameters βand ˙
Mrecovered from dataset C.
This was to be expected. In the case of the wind parameters
the precision with which information about these parameters
can be recovered from the line profiles decreases with decreas-
ing wind density (e.g. Puls et al. 1996). Still, the precision with
which the wind parameters are recovered for the weak wind
data set C, is remarkable.
A similar reasoning applies for the microturbulent veloc-
ity recovered from data set A. For low values of the micro-
turbulence, i.e. vturb <vth, thermal broadening will dominate
over broadening due to microturbulence. This decreases the
precision with which this parameter can be recovered from the
line profiles. Realizing that in case of this dataset for helium
vth ≈14 km s−1, again, the precision with which vturb is recov-
ered, is impressive.
To illustrate how quickly and how well the input parame-
ters are recovered Fig. 1 shows the evolution of the fit parame-
ters during the fit of synthetic dataset B. Also shown, as a grey
dashed line, is the fitness of the best fitting model found, dur-
ing the run. This fitness is normalized with respect to the fitness
of the model used to create the synthetic data (the data being
the combination of this model and noise). Note that the final
maximum normalized fitness found by the method exceeds 1.0,
which is due to the added noise allowing a further fine tuning
of the parameters by the GA based optimization. As can be
seen in this figure the method modifies multiple parameters si-
multaneously to produce a better fit. This allows for an efficient
exploration of parameter space and, more importantly, it allows
for the method to actually find the global optimum.
In the case of dataset B finding the global optimum required
only a few tens of generations (∼30). For the other two datasets
all save one parameter were well established within this num-
ber of generations. To establish the very low value of vturb in
dataset A and the very low ˙
Min dataset C required ∼100 gen-
erations. We will adopt 150 generations to fit the spectra in
Sect. 3. One reason, obviously, is that to safeguard that the
global optimum is found. A second reason, however, is that it
assures that the errors on the model parameters that we deter-
mine are meaningful (i.e. it assures that the error on the error is
modest).
We consider doing such a formal test as performed above
as part of the analysis of a set of observed spectra, as the exact
number of generations required is, in principle, a function of
e.g. the signal-to-noise ratio and the spectral resolution. Also,
special circumstances may play a role, such as potential nebular
contamination (in which case the impact of removing the line
cores from the fit procedure needs to be assessed).
3. Spectral analysis of early-type stars
In this section we apply our fitting method to seven stars in the
open cluster Cyg OB2, previously analysed by Herrero et al.
(2002) and five “standard” early-type stars, 10 Lac, τSco,
ζOph, HD 15629 and HD 217086, previously analysed by var-
ious authors.
3.1. Description of the data
Table 2 lists the basic properties of the data used for the analy-
sis. All spectra studied have a S/N of at least 100. The spectral
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 7
37
38
39
0 25 50 75
0.7
0.8
0.9
1.0
Teff [kK]
3.4
3.5
3.6
0 25 50 75
0.7
0.8
0.9
1.0
logg [cm s-2]
4.0
5.0
6.0
7.0
0 25 50 75
0.7
0.8
0.9
1.0
dM/dt [10-6 Msun yr-1]
0.8
1.0
1.2
0 25 50 75
0.7
0.8
0.9
1.0
β
0.08
0.10
0.12
0.14
0 25 50 75
0.7
0.8
0.9
1.0
YHe
generation
0.0
4.0
8.0
12.0
0 25 50 75
0.7
0.8
0.9
1.0
vturb [km s-1]
generation
Fig.1. Evolution of the best fitting model parameters for formal test B. From the 200 generation run only the first 75 generations
are shown. For this specific data set the location of the global optimum is found within 50 generations. This is indicated by the
highest fitness found during the run, which is shown as a grey dashed line and is scaled to the right vertical axis. The fitness is
normalized with respect to the fitness of the model used to create the synthetic data (the data being this model plus noise).
resolution of the data in the blue (regions between ∼4000 and
∼5000 Å) and the red (region around Hα) is given in Tab. 2.
The optical spectra of the stars in Cyg OB2 were obtained
by Herrero et al. (1999) and Herrero et al. (2000). Absolute
visual magnitudes of the Cyg OB2 objects were adopted
from Massey & Thompson (1991), and correspond to a dis-
tance modulus of 11.2m. Note that for object #8A Tab. 7 in
Massey & Thompson contains an incorrect V0value of 4.08m.
This should have been 4.26mconform the absorption given in
this table and the visual magnitude in their Tab. 2. For vrsin i
values determined by Herrero et al. (2002) are used, with the
exception of objects #8A and #10. For these we found that
the Heand metal lines are somewhat better reproduced if
we adopt vrsin ithat are higher by ∼35% and ∼10%, respec-
tively. Terminal flow velocities of the wind have been obtained
from UV spectra obtained with Hubble Space Telescope (cf.
Herrero et al. 2001). Data of HD 15629, HD 217086 and ζOph
are from Herrero et al. (1992) and Herrero (1993). For MV,v∞
and vrsin ivalues given by Repolust et al. (2004) are adopted.
The distances to these objects are based on spectroscopic paral-
laxes, except for ζOph which has a reliable Hipparcos distance
(Schr¨oder et al. 2004).
The spectrum of 10 Lac was obtained by Herrero et al.
(2002). The absolute visual magnitude of this star is from
Herrero et al. (1992). For v∞we adopted the minimum value
which is approximately equal to the escape velocity at the stel-
lar surface of this object. For the projected rotational velocity
we adopt 35 km s−1. The blue spectrum of τSco is from Kilian
(1992). The red region around Hαwas observed by Zaal et al.
(1999). For τSco we also adopt the Hipparcos distance. This
distance results in an absolute visual magnitude which is rather
large for the spectral type of this object, but is in between the
MVadopted by Kilian (1992) and Humphreys (1978). For the
projected rotational velocity a value of 5 km s−1was adopted.
3.2. Lines selected for fitting and weighting scheme
For the analysis will fit the hydrogen and helium spec-
trum of the investigated objects. Depending on the wavelength
range of the available data, these lines comprise for hydrogen
the Balmer lines Hα, Hβ, Hγand Hδ; for He the singlet lines at
4387 and 4922 Å, the He triplet lines at 4026, which is blended
with He , 4471 and 4713 Å; and finally for He the lines at
4200, 4541 and 4686 Å.
For an efficient and reliable use of the automated method
we have to incorporate into it the expertise that we have devel-
oped in the analysis of OB stars. The method has to take into
account that some lines may be blended or that they cannot
8 M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
Table 2. Basic parameters of the early type stars studied here. Spectral types are taken from Massey & Thompson (1991),
Walborn (1972, 1973) and Conti & Alschuler (1971). Blue and red resolution, respectively, correspond to the region between
∼4000 and ∼5000 Å and the region around Hα.
Star Spectral MVBlue Red vrsin i v∞
Type resolution [Å] resolution [Å] [km s−1] [kms−1]
Cyg OB2 #7 O3 If∗−5.91 0.6 0.8 105 3080
Cyg OB2 #11 O5 If+−6.51 1.3 0.8 120 2300
Cyg OB2 #8C O5 If −5.61 1.3 0.8 145 2650
Cyg OB2 #8A O5.5 I(f) −6.91 0.6 0.8 130 2650
Cyg OB2 #4 O7 III((f)) −5.44 1.3 0.8 125 2550
Cyg OB2 #10 O9.5 I −6.86 0.6 0.8 95 1650
Cyg OB2 #2 B1 I −4.64 0.6 0.8 50 1250
HD15629 O5 V((f)) −5.50 0.6 0.8 90 3200
HD217086 O7 Vn −4.50 0.6 0.8 350 2550
10 Lac O9 V −4.40 0.6 0.6 35 1140
ζOph O9 V −4.35 0.6 0.8 400 1550
τSco B0.2 V −3.10 0.2 0.2 5 2000
Table 3. Line weighting scheme adopted for different spectral
types and luminosity classes for the objects fitted in this paper.
Late, mid and early spectral type correspond to, respectively,
[O2-O5.5], [O6-O7.5] and [O8-B1]. The weights are imple-
mented in the fitness definition according to Eq. (6) and have
values of 1.0, 0.5 and 0.25 in case of h, m and l, respectively.
Dwarfs Super Giants
Late Mid Early Late Mid Early
H Balmer h h h h h h
He singlets h l l h l l
He 4026 h h h h h h
He 4471 h h h l m h
He 4713 h h h h h h
He 4686 h m m m m m
He 4541 h h h h h h
He 4200 m m m m m m
be completely reproduced by the model atmosphere code for
whatever reason For example, the so-called “generalized dilu-
tion effect” Voels et al. (1989), present in the He λ4471 line in
late type supergiants, that is still lacking an explanation.
To that end we have divided the stars in two classes
(“dwarfs” and “supergiants”, following their luminosity class
classification1), and three groups in each class (following spec-
tral types). We have then a total of six stellar groups, and
1For the one giant in our sample, Cyg OB2 #4, we have adopted
the line weighting scheme for dwarfs.
have assigned the spectral lines different weights depending
on their behaviour in each stellar group. This behaviour rep-
resents the expertise from years of “by eye” data analysis that
is being translated to the method. Three different weights are
assigned to each line: high, to lines very reliable for the anal-
ysis; medium, and low. The implementation of these weights
into the fitness definition is given by
F≡
N
X
i
wiχ2
red,i
−1
,(6)
where the parameter wicorresponds to the weight of a specific
line.
Table 2 gives the weights assigned to each line in each stel-
lar group. We will only briefly comment on the low or medium
weights. He singlets are assigned a low weight for mid-type
stars because of the singlet differential behaviour found be-
tween and (Puls et al. 2005), while they are
very weak for early-type stars. In these two cases therefore we
prefer to rely on the triplet He λ4471 line. To this line, how-
ever, a low weight is assigned at late-type Supergiants because
of the above mentioned dilution effect.
He λ4686 is only assigned a medium weight (except for
late type dwarfs), as this line is not always completely consis-
tent with the mass-loss rates derived from Halpha. He λ4200
is sometimes blended with N λ4200, and sometimes it is not
completely consistent with the rest of the He lines. He and
He lines at 4026 Å do overlap, but for both lines we find a
consistent behaviour.
The highest weight is therefore given to the Balmer lines
plus the He λ4541 and the He /He 4026 lines, which define
the He ionization balance with He λ4471 or the singlet He
lines. Note however that, as discussed above, all lines fit simul-
taneously in a satisfactory way for our best fitting models.
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 9
Table 4. Results obtained for the investigated early type stars using GA optimized spectral fits. The spectra were fitted by
evolving a population of 72 models over a course of 150 generations. Spectroscopic masses Msare calculated with the
gravities corrected for centrifugal acceleration log gc. Evolutionary masses Mev are from Schaller et al. (1992). The error bars on
the derived parameters are given in Tab. 5 and are discussed in Sect. 4.
Star Tefflog glog gcR⋆log L⋆YHe vturb ˙
MβMsMev
[kK] [cm s−2] [cm s−2] [R⊙] [L⊙] [km s−1] [M⊙yr−1] [M⊙] [M⊙]
Cyg OB2 #7 45.8 3.93 3.94 14.4 5.91 0.21 19.9 9.98·10−60.77 65.1 67.8
Cyg OB2 #11 36.5 3.62 3.63 22.1 5.89 0.10 19.8 7.36·10−61.03 75.9 55.6
Cyg OB2 #8C 41.8 3.73 3.74 13.3 5.69 0.13 0.5 3.37·10−60.85 36.0 49.2
Cyg OB2 #8A 38.2 3.56 3.57 25.6 6.10 0.14 18.3 1.04·10−50.74 89.0 74.4
Cyg OB2 #4 34.9 3.50 3.52 13.7 5.40 0.10 18.9 8.39·10−71.16 22.4 32.5
Cyg OB2 #10 29.7 3.23 3.24 29.9 5.79 0.08 17.0 2.63·10−61.05 56.0 45.9
Cyg OB2 #2 28.7 3.56 3.57 11.3 4.88 0.08 16.5 1.63·10−70.801) 17.0 18.7
HD 15629 42.0 3.81 3.82 12.6 5.64 0.10 8.6 9.28·10−71.18 37.8 47.4
HD 217086 38.1 3.91 4.01 8.30 5.11 0.09 17.1 2.09·10−71.27 25.7 28.5
10 Lac 36.0 4.03 4.03 8.27 5.01 0.09 15.5 6.06·10−80.801) 26.9 24.9
ζOph 32.1 3.62 3.83 8.9 4.88 0.11 19.7 1.43·10−70.801) 19.5 20.3
τSco 31.9 4.15 4.15 5.2 4.39 0.12 10.8 6.14·10−80.801) 13.7 16.0
1) assumed fixed value
3.3. Fits and comments on the individual analysis
In the following we will present the fits that were obtained by
the automated method for our sample of 12 early type stars,
and comment on the individual analysis of the objects. Listed
in Tab. 4 are the values determined for the six free parameters
investigated and quantities derived from these.
3.3.1. Analysis of the Cyg OB2 stars
The Cyg OB2 objects studied here were previously analysed
by Herrero et al. (2002, hereafter HPN). We opted to reanal-
yse these stars (to test our method) as these stars have equal
distances and have been analysed in a homogeneous way us-
ing (an earlier version of) the same model atmosphere code. In
Sect. 5 we will systematically compare our results with those
obtained by HPN. Here, we will incidentally discuss the agree-
ment if this turns out to be relatively poor or if the absolute
value of a parameter seems unexpected, and we wanted to test
possible causes for the discrepancy.
Cyg OB2 #7 The best fit obtained with our automated fitting
method for Cyg OB2 #7 is shown in Fig. 2. For all hydrogen
lines fitted, including Hδnot shown here, and all He lines the
fits are of very good quality. Note that giventhe noise level the
fits of the He lines are also acceptable.
Interesting to mention is the manner in which the He and
He blend at 4026 Å is fitted. At first sight, i.e. “by eye”, it
seems that the fit is of poor quality, as the line wings of the
synthetic profile runs through “features” which might be at-
tributed to blends of weak photospheric metal lines. However,
the broadest of these features have a half maximum width of
∼70 km s−1, which is much smaller than the projected rota-
tional velocity of 105 km s−1. Consequently, these features are
dominated by pure noise.
Compared to the investigation of HPN we have partial
agreement between the derived parameters. The mass loss rate,
Teffand to a lesser degree βagree very well. For log gand
the helium abundance we find, however, large differences. The
log gvalue obtained here is ∼0.2 dex larger, which results in
a spectroscopic mass of 65.1 M⊙. A value which is in good
agreement with the evolutionary mass of 67.8 M⊙.
The helium abundance needed to fit this object is 0.21,
which is considerably lower than the value obtained by HPN,
who found an abundance ratio of 0.31. This large value still
corresponds to a strong helium surface enrichment. An inter-
esting question we need to address, is whether this is a real
enrichment and not an artifact that is attributable to a degen-
eracy effect of Teffand YHe. The latter can be the case, as no
He lines are present in the optical spectrum of Cyg OB2 #7.
This issue can be resolved with our fitting method by refitting
the spectrum with a helium abundance fixed at a lower value
than previously obtained. If Teffand YHe are truly degenerate
this would again yield a good fit, however for a different Teff.
Shown as dotted lines in Fig. 2 are the results of refitting
Cyg OB2 #7 with a helium abundance fixed at the solar value.
For this lower YHe aTeffthat is lower by ∼2.1 kK was obtained.
This was to be expected as for this temperature regime He
is the dominant ionization stage. When consequently YHe is re-
duced a reduction of the temperature is required to fit the He
lines. The reduction of Teffobtained is the maximum for which
still a good fit of the hydrogen lines is possible and the He
lines do not become too strong. More importantly, in Fig. 2 it
is shown that even with this large reduction of Teffthe He
10 M.R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
0.70
0.80
0.90
1.00
4330 4340 4350
Hγ
0.70
0.80
0.90
1.00
4850 4860 4870
Hβ
0.95
1.00
1.05
1.10
6520 6560 6600
Hα
0.90
0.95
1.00
4020 4025 4030
HeI + HeII 4026
0.98
1.00
1.02
4384 4388 4392
HeI 4387
0.98
1.00
1.02
4468 4472 4476
HeI 4471
0.98
1.00
1.02
4915 4920 4925
HeI 4922
0.85
0.90
0.95
1.00
4190 4200 4210
HeII 4200
0.80
0.90
1.00
4530 4540 4550
HeII 4541
1.00
1.05
1.10
1.15
4680 4690 4700
HeII 4686
Fig.2. Comparison of the observed line profiles of Cyg OB2 #7 with the best fit obtained by the automated fitting method
(dashed lines). Note that the He line at 6527.1 Å is not included in the fit and, therefore, disregarded by the automated method.
Horizontal axis gives the wavelength in Å. Vertical axises give the continuum normalized flux and are scaled differently for each
line. In this figure the dotted lines correspond to a fit obtained for a helium abundance fixed at 0.1. See text for further comments.
0.70
0.80
0.90
1.00
4330 4340 4350
Hγ
0.80
0.90
1.00
4850 4860 4870
Hβ
0.90
1.00
1.10
1.20
6520 6560 6600
Hα
0.98
0.99
1.00
1.01
4384 4388 4392
HeI 4387
0.92
0.96
1.00
4468 4472 4476
HeI 4471
0.98
0.99
1.00
1.01
4915 4920 4925
HeI 4922
0.90
0.95
1.00
4190 4200 4210
HeII 4200
0.85
0.90
0.95
1.00
4530 4540 4550
HeII 4541
1.00
1.05
1.10
1.15
4680 4690 4700
HeII 4686
Fig.3. Same as Fig. 2, however for Cyg OB2 #11
lines cannot be fitted. This implies that Teffand YHe are not
degenerate and the obtained helium enrichment is real.
Cyg OB2 #11 Figure 3 shows the fit to Cyg OB2 #11. In gen-
eral all lines are reproduced correctly. There is a slight un-
der prediction of the cores of Hγand He λ4541, a problem
that was also pointed out by Herrero et al. (1992) and HPN.
Possibly this is due to too much filling in of the predicted
profiles by wind emission. Part of the He λ4541 discrep-
ancy might be related to problems in the theoretical broadening
functions (see Repolust et al. 2005).
The parameters obtained for this object, with exception of
˙
M, are in agreement with the parameters derived by HPN. With
our automated method a mass loss rate lower by ∼0.1 dex was
obtained. Note that due to this lower valuethe behaviour of this
object in terms of its modified wind momentum (cf. Sect. 5.4)
is in better accord with that of the bulk of the stars investigated
in this paper.
Cyg OB2 #8C The best fit for Cyg OB2 #8C is shown in
Fig. 4. Again, with exception of the mass loss rate, the param-
eters we obtain for this object are in good agreement with the
findings of HPN. We do find a small helium abundance en-
hancement, whereas HPN found a solar value.
To fit the P Cygni type profile of He at 4686 Å, the au-
tomated method used a ˙
Mwhich, compared to these authors,
was higher by approximately 0.15 dex. This higher value for
the mass loss rate results in a Hαprofile which, at first sight,
looks to be filled in too much by wind emission. To assess
whether this could correspond to a significant overestimation of
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 11
0.70
0.80
0.90
1.00
4330 4340 4350
Hγ
0.70
0.80
0.90
1.00
4850 4860 4870
Hβ
0.85
0.90
0.95
1.00
6540 6560 6580
Hα
0.99
1.00
1.01
4384 4388 4392
HeI 4387
0.96
0.98
1.00
4468 4472 4476
HeI 4471
0.99
1.00
1.01
4915 4920 4925
HeI 4922
0.90
0.95
1.00
4190 4200 4210
HeII 4200
0.85
0.90
0.95
1.00
4530 4540 4550
HeII 4541
0.95
1.00
1.05
4680 4690 4700
HeII 4686
Fig.4. Same as Fig. 2, however for Cyg OB2 #8C. Shown with a dotted line for Hαand He λ4686 are the line profiles of a
model with a 0.05 dex lower ˙
M, which “by eye” fits the core of Hα. See text for further comments.
0.70
0.80
0.90
1.00
4090 4100 4110
Hδ
0.70
0.80
0.90
1.00
4330 4340 4350
Hγ
0.70
0.80
0.90
1.00
4850 4860 4870
Hβ
0.90
0.95
1.00
1.05
6530 6560 6590
Hα
0.98
1.00
1.02
4384 4388 4392
HeI 4387
0.92
0.96
1.00
4468 4472 4476
HeI 4471
0.98
1.00
1.02
4915 4920 4925
HeI 4922
0.85
0.90
0.95
1.00
4190 4200 4210
HeII 4200
0.80
0.90
1.00
4530 4540 4550
HeII 4541
0.95
1.00
1.05
4680 4690 4700
HeII 4686
Fig.5. Same as Fig. 2, however for Cyg OB2 #8A. The dotted lines correspond to a model with a ˙
Mhigher by 0.04 dex. This
mass loss rate was obtained by fitting the best fit model, found by the automated method, “by eye” to the Hαcore. Even though
the fit obtained with the higher ˙
Mresults in a fit of Hαwhich is more pleasing to the eye in the line core, this higher mass loss
rate does not describe this object the best. This can be seen best from the reduced fit quality of the other hydrogen Balmer lines
and the severe mismatch of He λ4471. See text for further comments.
the mass loss rate, we lowered ˙
Min the best fit model by hand
until the core of Hαwas fitted. In Fig. 4 the resulting line pro-
files are shown as a dotted line for Hαand He λ4686, which
for this fit are the lines which visibly reacted to the change in
mass loss rate. To obtain this fit “by eye” of the Hαcore, a
reduction of ˙
Mwith merely 0.05 dex was required, showing
that the mass loss rate was not overestimated by the automated
method. Note that for this lower mass loss rate the fit of the
He λ4686 becomes significantly poorer.
Cyg OB2 #8A De Becker et al. (2004) report this to be a O6 I
and O5.5 III binary system, therefore the derived parameters,
in particular the spectroscopically determined mass, should be
taken with care. However, as this paper also aims to test auto-
mated fitting we did pursue the comparison of this object with
HPN, who also treated the system assuming it to be a single
star.
We obtained a good fit for all lines except for the problem-
atic He λ4686 line. The best fit is shown in Fig. 5. Again the
Hαcore is not fitted perfectly. To determine how significant this
small discrepancy is, we fitted the Hαcore in a similar manner
as for Cyg OB2 #8C. To obtain a good fit “by eye” we find that
˙
Mhas to be increased by 0.04 dex, indicating the extreme sen-
sitivity of Hαto ˙
Min this regime. The profiles corresponding
to the increased mass loss rate model are shown in Fig. 5 as
dotted lines. It is clear that not only the “classical” wind lines
12 M.R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
0.70
0.80
0.90
1.00
4330 4340 4350
Hγ
0.70
0.80
0.90
1.00
4850 4860 4870
Hβ
0.80
0.90
1.00
6540 6560 6580
Hα
0.94
0.96
0.98
1.00
4384 4388 4392
HeI 4387
0.85
0.90
0.95
1.00
4468 4472 4476
HeI 4471
0.94
0.96
0.98
1.00
4708 4712 4716
HeI 4713
0.92
0.96
1.00
4918 4922 4926
HeI 4922
0.90
0.95
1.00
4190 4200 4210
HeII 4200
0.85
0.90
0.95
1.00
4530 4540 4550
HeII 4541
0.90
0.95
1.00
4675 4685 4695
HeII 4686
Fig.6. Same as Fig. 2, however for Cyg OB2 #4.
0.70
0.80
0.90
1.00
4330 4340 4350
Hγ
0.70
0.80
0.90
1.00
4850 4860 4870
Hβ
0.92
0.96
1.00
1.04
6540 6560 6580
Hα
0.85
0.90
0.95
1.00
4384 4388 4392
HeI 4387
0.70
0.80
0.90
1.00
4468 4472 4476
HeI 4471
0.90
0.95
1.00
4710 4713 4716
HeI 4713
0.80
0.90
1.00
4919 4922 4925
HeI 4922
0.94
0.97
1.00
4195 4200 4205
HeII 4200
0.92
0.96
1.00
4538 4542 4546
HeII 4541
0.90
0.95
1.00
4680 4688 4696
HeII 4686
Fig. 7. Same as Fig. 2, however for Cyg OB2 #10. The emission feature in the core of Hαwas not included in the fit. A subsequent
test which did include this feature in the fit yielded the same parameters except for a small increase of ˙
Mwith 0.04 dex.
react strongly to ˙
M. All synthetic hydrogen Balmer line profiles
show significant filling in due to wind emission for an increased
mass loss, deteriorating the fit quality. Also the He λ4471 line
shows a decrease in core strength which is comparable to the
decrease in the Hαcore. This reconfirms that in order to self-
consistently determine the mass loss rate all lines need to be
fitted simultaneously. Therefore, a small discrepancy in the Hα
core between the observed and synthetic line profile should not
be considered a decisive reason to reject a fit.
Except for YHe the obtained parameters agree with the re-
sults of HPN within the errors given by these authors. Similar
to Cyg OB2 #8C we find a small helium enhancement.
Cyg OB2 #4 The final fit to the spectrum of Cyg OB2 #4 is
presented in Fig. 6. We obtained good fits for all lines, with
exception of the helium singlet at 4922 Å, for which the core is
predicted too strong. However, recall that for this spectral type
we assigned a relatively low weight to this line, for reasons
explained in Sect. 3.2.
The parameters obtained from the fit agree well with the
values of HPN, with exception of β, for which we find a value
higher by ∼0.2. Note that HPN used a fixed value for βto ob-
tain their fit, whereas in this case the automated method self-
consistently derived this parameter.
The spectroscopic mass implied by the obtained log g
value is significantly smaller than the evolutionary mass of
Cyg OB2 #4. However, within the error bars (Sect. 4) the two
masses agree with each other.
Cyg OB2 #10 In the final fit for this object, shown in Fig. 7,
there are two problematic lines. First, for the He λ4686 line
the core is predicted too strong. Even though compared to HPN
the situation has improved considerably, the current version
of still has difficulties predicting this line. Second,
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 13
0.60
0.80
1.00
4330 4340 4350
Hγ
0.60
0.80
1.00
4850 4860 4870
Hβ
0.60
0.80
1.00
6540 6560 6580
Hα
0.70
0.80
0.90
1.00
4384 4388 4392
HeI 4387
0.60
0.80
1.00
4468 4472 4476
HeI 4471
0.80
0.90
1.00
4710 4713 4716
HeI 4713
0.70
0.80
0.90
1.00
4919 4922 4925
HeI 4922
0.96
0.98
1.00
1.02
4190 4200 4210
HeII 4200
0.97
0.98
0.99
1.00
4530 4540 4550
HeII 4541
0.90
0.95
1.00
4680 4685 4690
HeII 4686
Fig.8. Same as Fig. 2, however for Cyg OB2 #2. Shown with dotted lines for Hαis the line profile of the best fit model with a ˙
M
lower by a factor of 3. See text for further comments.
the predicted He λ4471 is too weak. Possibly this is con-
nected to the generalized dilution effect, for which we refer to
Repolust et al. (2004) for a recent discussion.
In Fig. 7 we also see that the Hαcore of Cyg OB2 #10
exhibits an emission feature. For this analysis we assumed that
is was nebular and, consequently, excluded it from the fit. To
test what the effect would be if this assumption was incorrect, a
fit was made with this feature included in the profile. It turned
out that the only parameter which was affected in this test was
˙
M, which showed a small increase of 0.04 dex.
Cyg OB2 #2 For Cyg OB2 #2 the automated method could
not self-consistently determine β. Therefore, we fixed its value
at a theoretically predicted β=0.8 (cf. Pauldrach et al. 1986).
In Fig. 8 the best fit is shown. We obtained good fits for all
lines. However, in the case of He λ4471 we do see a small
under prediction of the forbidden component at 4469 Å, which
is likely related to incorrect line-broadening functions.
For log gand ˙
Mthe obtained fit parameters differ consider-
ably from the findings of HPN. We first focus on mass loss for
which we obtain the relatively low rate of 1.63 ×10−7M⊙yr−1
with an error bar in the logarithm of this value of −0.15 and
+0.12 dex (see Tab. 5), given the quoted value of β. Our ˙
M
value is approximately a factor two higher than the mass loss
rate obtained by HPN. These authors noted that it was not pos-
sible to well constrain the mass loss rate of such a weak wind.
Given the relatively modest errors indicated by our automated
fitting method, we conclude that at least in principle our tech-
nique allows to determine mass loss rates of winds as weak as
that of Cyg OB2 #2. We have added the phrase “in principle”
as it assumes the notion of beta and a very reliable continuum
normalization, which is in this is case different from the one
used by HPN for the He 4541, He 4686 and Hαlines. If this
can not be assured, then systematic errors may dominate over
the characteristic fitting error and the mass loss may be much
less well constrained. Assuming the continuum location to be
reliable, we illustrate the sensitivity of the spectrum to mass
loss rates of ∼10−7M⊙yr−1by reducing the mass loss by a fac-
tor of three. The Hαprofile of this reduced mass loss model is
shown in Fig. 8 as a dotted line. Comparison of these two cases
shows that for winds of order 10−7M⊙yr−1the line still contain
considerable ˙
Minformation. Interestingly, if we would not take
into consideration the line core of Hαin our fitting method, we
still recover the quoted mass loss to within three percent. We
note that for our higher mass loss this object appears to behave
well in the wind momentum luminosity relation (see Sect. 5.4),
whereas HPN signal a discrepancy when using their estimated
˙
Mvalue.
The log gvalue obtained in this study is 0.36 dex larger than
the value obtained “by eye” by HPN. Judging from the very
good fits obtained, there is no indication that the automated fit
overestimated the gravity. The spectroscopic mass of 17.0 M⊙
implied by the larger log gvalue, is also in good agreement
with the evolutionary mass of Cyg OB2 #2, which is 18.7 M⊙.
3.3.2. Analysis of well studied dwarf OB-stars
We have also reanalysed five well known and well studied
dwarf OB-stars, sampling the range of O spectral sub-types,
in order to probe a part of parameter space that is not well cov-
ered by the Cyg OB2 stars. HD 217086 and ζOph, for instance,
are fast rotators with vrsin i=350 and 400 km s−1respectively
(see also Tab. 2). 10 Lac is a slow rotator, and τSco is a very
slow rotator. The latter two stars also feature very low mass loss
rates, moreover, the actual ˙
Mvalues of these stars are much de-
bated (see Martins et al. 2004). HD 15629 is selected because
it appears relatively normal.
HD15629 Apart from a slight over-prediction of the core
strength in He λ4200, a very good fit was obtained for this
object. The final fit is presented in Fig. 9. This object has re-
cently been studied by Repolust et al. (2004, hereafter RPH).
14 M.R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
0.60
0.80
1.00
4090 4100 4110
Hδ
0.60
0.80
1.00
4330 4340 4350
Hγ
0.60
0.80
1.00
4850 4860 4870
Hβ
0.80
0.90
1.00
6540 6560 6580
Hα
0.99
1.00
1.01
4384 4388 4392
HeI 4387
0.92
0.96
1.00
4467 4471 4475
HeI 4471
0.98
0.99
1.00
1.01
4917 4921 4925
HeI 4922
0.85
0.90
0.95
1.00
4190 4200 4210
HeII 4200
0.80
0.90
1.00
4530 4540 4550
HeII 4541
0.80
0.90
1.00
4675 4685 4695
HeII 4686
Fig.9. Same as Fig. 2, however for HD 15629.
0.80
0.90
1.00
4080 4100 4120
Hδ
0.80
0.90
1.00
4320 4340 4360
Hγ
0.80
0.90
1.00
4840 4860 4880
Hβ
0.80
0.90
1.00
6540 6560 6580
Hα
0.98
1.00
4380 4390 4400
HeI 4387
0.94
0.97
1.00
4463 4471 4479
HeI 4471
0.96
0.98
1.00
4914 4922 4930
HeI 4922
0.92
0.96
1.00
4190 4200 4210
HeII 4200
0.92
0.96
1.00
4530 4540 4550
HeII 4541
0.90
0.95
1.00
4675 4685 4695
HeII 4686
Fig.10. Same as Fig. 2, however for HD 217086.
Compared to the parameters obtained by these authors, we find
good agreement except for Teff,˙
Mand β. Note that in contrast
to this study we do not find a helium deficiency. However, the
difference of 0.02 with respect to the solar value obtained here
is within the error quoted by RPH.
The difference in wind parameters can be explained by the
value of β=0.8 assumed by RPH. Our self-consistently de-
rived value for β=1.18. As the effect of βon the spectrum is
connected to the mass loss rate through the velocity law and the
continuity equation, the lower ˙
Mobtained with the automated
method is explained.
The 1.5 kK increase of Teffcompared to RPH can be at-
tributed to the improved fit quality and the increase in log gof
0.1 dex. An increase in loggimplies an increase in electron
density, resulting in an increase in the recombination rate. The
strength of both the He and He lines depend on this rate, as
the involved levels are mainly populated through recombina-
tion. Consequently, as He is the dominant ionization stage in
the atmosphere of HD 15629 the strength of the Heand He
lines will increase when the recombination rate increases. To
compensate for this increase in line strength an increase in Teff,
decreasing the ionization fractions of He and He , is neces-
sary.
HD217086 With a projected rotational velocity of 350 km s−1
this object can be considered to be a fast rotator, and our analy-
sis of this object will show how well the automated method can
handle large vrsin i. In Fig. 10 the best fit obtained with our
method is presented. We find that the large projected rotational
velocity does not pose any problem for the method, i.e. the fit
quality of all the lines fitted is very good.
With respect to the obtained parameters, again, these can
be compared to the work of RPH. In this comparison we find
considerable differences for Teffand loggand a small differ-
ence for YHe. The effective temperature found by the automated
method is 2.1 kK higher. This is a significant increase, but
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 15
0.60
0.80
1.00
4320 4340 4360
Hγ
0.60
0.80
1.00
4840 4860 4880
Hβ
0.60
0.80
1.00
6540 6560 6580
Hα
0.80
0.90
1.00
4384 4388 4392
HeI 4387
0.60
0.80
1.00
4466 4471 4476
HeI 4471
0.80
0.90
1.00
4711 4713 4715
HeI 4713
0.70
0.80
0.90
1.00
4919 4922 4925
HeI 4922
0.85
0.90
0.95
1.00
4190 4200 4210
HeII 4200
0.85
0.90
0.95
1.00
4530 4540 4550
HeII 4541
0.60
0.80
1.00
4680 4685 4690
HeII 4686
Fig.11. Same as Fig. 2, however for 10 Lac.
0.80
0.90
1.00
4325 4340 4355
Hγ
0.80
0.90
1.00
4845 4860 4875
Hβ
0.80
0.90
1.00
6540 6560 6580
Hα
0.94
0.97
1.00
4380 4390 4400
HeI 4387
0.90
0.95
1.00
4460 4470 4480
HeI 4471
0.96
0.98
1.00
4704 4712 4720
HeI 4713
0.92
0.96
1.00
4910 4920 4930
HeI 4922
0.96
0.98
1.00
4185 4200 4215
HeII 4200
0.96
0.98
1.00
4530 4540 4550
HeII 4541
0.94
0.96
0.98
1.00
4675 4685 4695
HeII 4686
Fig.12. Same as Fig. 2, however for ζOph.
when the log gvalue obtained here is considered, this can be
explained in a similar manner as the Teffincrease of HD 15629.
The best fit is obtained with a log gvalue that is 0.29 dex
higher than the value from RPH. Judging from the line profiles
in Fig. 10 there is no evidence for an overestimation of logg.
This higher log gremoves the discrepancy with the calibration
of Markova et al. (2004) found by RPH (see Fig. 17 in RPH).
We also note that, similar to Cyg OB2 #2, the increased log g
implies a spectroscopic mass which agrees well with the evo-
lutionary mass of HD 217086 (cf. Tab. 4). This is not the case
for the value determined by RPH, which points to a clear dis-
crepancy.
The considerable helium abundance enhancement found by
RPH is not reproduced by the automated method. Even though
this object is a rapid rotator, our fit indicates a normal, i.e. solar,
helium abundance.
10 Lac Like in the case of Cyg OB2 #2, and for the remaining
objects, the wind is too weak to self-consistently determine β.
Therefore, again a value of β=0.8 was assumed.
The photospheric parameters obtained for 10 Lac agree
very well with the results of HPN. The best fit to the ob-
served spectrum is shown in Fig. 11. Whereas HPN find that the
mass loss rate cannot be constrained and only an upper limit of
10−8M⊙yr−1is found, the automated method was able to self-
consistently determine ˙
Mat 6×10−8M⊙yr−1, though with large
error bars (see Tab. 5). Our error bar indicates that ˙
Mmay be
an order of magnitude lower, i.e. it may still be consistent with
the HPN result.
Various other authors have determined the mass loss rate
of 10 Lac using different methods. These determinations range
from up to 2 ×10−7(Howarth & Prinja 1989) down to 2 ×
10−9M⊙yr−1(Martins et al. 2004). Consequently, compared
to these independent determinations no conclusive answer can
be given to the question whether the ˙
Mderived from the opti-
16 M.R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
0.60
0.80
1.00
4325 4340 4355
Hγ
0.60
0.80
1.00
4845 4860 4875
Hβ
0.70
0.80
0.90
1.00
6540 6560 6580
Hα
0.60
0.80
1.00
4384 4388 4392
HeI 4387
0.60
0.80
1.00
4468 4472 4476
HeI 4471
0.70
0.80
0.90
1.00
4711 4713 4715
HeI 4713
0.60
0.80
1.00
4918 4922 4926
HeI 4922
0.90
0.95
1.00
4195 4200 4205
HeII 4200
0.94
0.97
1.00
4530 4540 4550
HeII 4541
0.80
0.90
1.00
4680 4685 4690
HeII 4686
Fig.13. Same as Fig. 2, however for τSco.
cal spectrum is correct. We conclude that the mass loss rate of
10 Lac is anomalously low when placed into context with the
other dwarfs stars studied here. For instance, the dwarfs ζOph
and HD 217086, which have luminosities that are, respectively,
lower and higher by ∼0.1 dex, both exhibit a mass loss rate
higher by several factors. In Sect. 5.4 we will discuss this fur-
ther in terms of the wind-momentum luminosity relation.
ζOph The large vrsin iof 400 km s−1was not a problem to
obtain a good fit. In Fig. 12 the best fit for ζOph is pre-
sented. With the exception of the helium abundance, the com-
parison with the results of RPH yields very good agreement.
Note that the mass loss rate obtained by these authors is an
upper limit, whereas in this study ˙
Mcould be derived self-
consistently. With respect to YHe we do not find any evidence
for a significant overabundance of helium, in agreement with
Villamariz & Herrero (2005).
τSco The best fit for τSco is presented in Fig. 13. All lines,
including Hδ, which is not shown here, are reproduced accu-
rately. The photospheric parameters we obtained can be com-
pared to the work of Sch ¨onberner et al. (1988) and Kilian et al.
(1991) who both studied τSco using plane parallel mod-
els. Kilian et al. found Teff=31.7 kK and log g=4.25, whereas
Sch¨onberner et al. obtained Teff=33.0 kK and log g=4.15. The
difference in Teffbetween the two studies is explained by the
fact that in the latter analysis no line blanketing was included
in the models. Therefore, we prefer to compare our Teffto the
former investigation, which agree very well. In terms of the
gravity we find good agreement with the second study. The
value obtained by Kilian et al. seems rather high. Given the al-
most perfect agreement between the synthetic line profiles and
the observations in Fig. 13, the reason for this discrepancy is
unclear. On a side note, more recently Repolust et al. (2005)
analysed the infrared spectrum of this object. Their findings do
confirm our lower value, but could not reproduce the enhanced
helium abundance we find, due to a lack of observed infrared
He lines.
In the recent literature the mass loss rate usually adopted
for τSco is 9 ×10−9M⊙yr−1, which is considerably smaller
than the 6.14 ×10−8M⊙yr−1obtained in this study. However,
the former mass loss rate is an average value determined by
de Jager et al. (1988), based on the mass loss rates indepen-
dently found by Gathier et al. (1981) and Hamann (1981).
Based on the UV resonance lines, these two studies, respec-
tively, determined ˙
Mto be 7.4×10−8and 1.3×10−9M⊙yr−1.
So, they differ by more than a factor of 50. The mass loss rate
obtained with the automated method is in reasonable agreement
with that obtained by Gathier et al.. Our higher value is also
supported by the study of the infrared spectrum of τSco by
Repolust et al. (2005) who find ˙
M≃2×10−8M⊙yr−1. Detailed
fitting of Brαwill likely clarify this issue.
4. Error analysis
Here we will introduce our method of estimating errors on the
parameters derived with the automated method. This method
is based on properties of the distribution of the fitnesses of the
models in parameter space, which may seem conceptually dif-
ferent from classical approaches of defining error bars (and in
a sense it is). However, we will demonstrate for the case of
10 Lac that our error definition is very comparable to what is
routinely done in fit diagram approaches.
4.1. Fit diagrams
In a fit diagram method the error bar on Teffand loggis de-
rived by investigating the simultaneous behaviour of these two
parameters. In panel aof Fig. 14 the fit diagram of 10 Lac is
presented adopting for all other parameters (save Teffand log g)
the best fit values obtained in our automated fitting. This dia-
gram was constructed by calculating a grid of models
in the Teff-log gplane, and evaluating for every line for every
Teffwhich model, i.e. log g, fits this line the best. The loca-
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 17
3.0
3.5
4.0
4.5
5.0
5.5
30 32 34 36 38 40 42
logg [cm s
−2
]
Teff [kK]
H
H
H
HeII 4200
HeII 4541
HeII 4686
HeI 4387
HeI 4471
HeI 4713
HeI 4922
0
100
200
300
400
500
30 32 34 36 38 40 42
Fitness
Teff [kK]
0
50
100
150
200
250
0.5 0.6 0.7 0.8 0.9 1
#models
Fitness
0
200
400
600
800
1000
1200
33 34 35 36 37 38
#models
Teff [kK]
α
β
γ
a b
dc
Fig.14. Panel a: fit diagram of Teffand log gfor 10 Lac. Panel b: Fitness as a function of Tefffor log g=4.0. Panel c: Fitness
distribution of the models calculated during the fitting run of the automated method. Panel d: Distribution of Teffin the models
located within the global optimum. The maximum variation of Teffwithin the global optimum, which corresponds to the error
estimate of this parameter, is ∼900 K.
tion where the resulting fit curves intersect, corresponds to the
best fit. This best fit yields Teff=36000 K and log g=4.0.
Note that this result was obtained without the use of our auto-
mated method. The error can now be estimated by estimating
the dispersion of the fit curves around this location. In panel a
of Fig. 14 this is indicated by a box around the best fit loca-
tion. The corresponding error estimates are 1000 K in Teffand
0.1 dex in log g.
The method described above cannot be applied to our au-
tomated fitting method due to two reasons. First, as we have
defined the fit quality according to Eq. (1), this definition of
fitness compresses the fit curves of all individual lines in the
fit diagram to a single curve. In Fig. 14 this curve is shown as
a thick dashed line. Although the curve runs through the best
fit point, no information about the dispersion of the solutions
around this point can be derived from it. The second reason
lies in the multidimensional character of the problem of line
fitting. If one would want to properly estimate the error tak-
ing this multidimensionality in to account, a fit diagram should
be constructed with a dimension equal to the number of free
parameters evaluated. In case of our fits this translates to the
construction of a six dimensional fit diagram.
4.2. Optimum width based error estimates
Even though we have argued that fit diagrams cannot be used
with our fitting method, it is possible to construct an error esti-
mate which is analogous to the use of these diagrams and does
take the multidimensionality of the problem into account. This
can be done by first realizing that the error box shown in Fig. 14
essentially is a measure of the width of the optimum in parame-
ter space, i.e. it defines the region in which models are located
which approximately have the same fit quality. This is illus-
trated in panel b. There we show the one-dimensional fitness
function in the Teff-log gplane for logg=4.0. Indicated with
dashed lines is the error in Teffestimated using the fit diagram
of 10 Lac. Confined between these lines is the region which
corresponds to the optimum as defined by the error box in panel
a. Consequently, the difference between maximum and mini-
mum fitness in this region defines the width of the optimum.
Returning to the general case, we can now invert the reason-
ing and state that the error estimate for a given parameter is
equal to the maximum variation of this parameter in the group
of best fitting models, i.e. the models located within the error
box. Consequently, in the automated fitting method by defining
a group of best fitting models, the error estimates for all free
parameters can be determined.
We define the group of best fitting models as the group
of models that lie within the global optimum. Put differently,
18 M.R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
Table 5. Error estimates for fit parameters obtained using the automated fitting method and parameters derived from these.
Denoted by are errors in vturb that reach up to the maximum allowed value of vturb and, therefore, are formally not defined.
Uncertainties in the fit parameters result from the optimum width based error estimates method. See text for details and discussion.
Star ∆Teff∆log gc∆R⋆∆log L⋆∆YHe ∆vturb ∆log ˙
M∆β∆Ms∆Mev
[kK] [cms−2] [R⊙] [L⊙] [km s−1] [M⊙yr−1] [M⊙] [M⊙]
Cyg OB2 #7 −1.0
+1.5−0.08
+0.06 ±0.7 ±0.07 −0.02
+0.03 −14.9
+ND −0.05
+0.03 −0.04
+0.09 −15
+12 −7
+7
Cyg OB2 #11 −0.6
+0.4−0.07
+0.13 ±1.1 ±0.05 −0.01
+0.03 −4.0
+ND −0.03
+0.06 −0.05
+0.02 −15
+27 −3
+4
Cyg OB2 #8C −1.3
+1.1−0.10
+0.14 ±0.7 ±0.07 −0.02
+0.04 −0.2
+10.9−0.07
+0.04 −0.05
+0.10 −10
+14 −4
+4
Cyg OB2 #8A −0.4
+1.7−0.05
+0.13 ±1.3 ±0.09 −0.04
+0.04 −17.7
+ND −0.07
+0.03 −0.04
+0.11 −15
+32 −10
+8
Cyg OB2 #4 −0.3
+1.5−0.04
+0.21 ±0.7 ±0.09 −0.02
+0.03 −3.0
+ND −0.10
+0.05 −0.05
+0.21 −3
+15 −3
+3
Cyg OB2 #10 −0.8
+1.0−0.12
+0.16 ±1.5 ±0.07 −0.02
+0.03 −7.0
+ND −0.13
+0.08 −0.15
+0.19 −19
+26 −4
+4
Cyg OB2 #2 −0.8
+1.2−0.14
+0.13 ±0.6 ±0.08 −0.01
+0.03 −2.3
+2.4−0.15
+0.12 -−7
+6−1
+2
HD 15629 −0.3
+0.7−0.05
+0.07 ±1.9 ±0.12 −0.01
+0.03 −8.4
+7.6−0.13
+0.10 −0.10
+0.27 −13
+14 −5
+7
HD 217086 −0.5
+0.9−0.08
+0.07 ±1.2 ±0.13 −0.02
+0.02 −4.9
+2.9−0.12
+0.18 −0.25
+0.16 −10
+10 −3
+3
10 Lac −0.9
+0.8−0.12
+0.13 ±1.7 ±0.17 −0.02
+0.02 −3.8
+4.1−0.98
+0.39 -−16
+16 −2
+4
ζOph −0.7
+0.7−0.05
+0.16 ±1.3 ±0.13 −0.02
+0.04 −6.2
+ND −0.28
+0.15 -−7
+11 −2
+2
τSco −0.8
+0.5−0.14
+0.09 ±0.5 ±0.09 −0.02
+0.04 −2.2
+2.4−0.99
+0.22 -−6
+4−1
+1
the width of the global optimum in terms of fitness, defines
the group of best fitting models. Identifying and, consequently,
measuring this width is facilitated by the nature of the GA, i.e.
selected reproduction, incorporated in our fitting method. Due
to this selected reproduction the exploration through parame-
ter space results in a mapping of this space in which regions
of high fit quality, i.e. the regions around local optima and the
global optimum, are sampled more intensively. Consequently,
if we would rank all models of all generations calculated dur-
ing a fitting run according to their fitness, the resulting distri-
bution will peak around the locations of the optima. In case of
the global optimum the width of this peak, starting from up to
the maximum fitness found, is, analogous to the width of the
error box used in a fit diagram, a direct measure of the width of
the optimum. Consequently, this width depends on the quality
of the data, i.e. it will be broader or narrower for, respectively,
low and high signal to noise, and on the degeneracy between
the fit parameters. Therefore, the error estimates of the individ-
ual parameters are equal to the maximum variations of these
parameters for all models contained in the peak corresponding
to the global optimum.
In panel cof Fig. 14 the distribution of the models accord-
ing to their fitness calculated during the fitting run of 10 Lac
using the automated method is shown. The fitnesses are nor-
malized with respect to the highest fitness and only the top half
of the distribution is shown. In this distribution two peaks are
clearly distinguishable. The most pronounced peak is located
at F≈0.9 and corresponds to the region around the global op-
timum. A second peak, corresponding to a region around a sec-
ondary optimum, is located at F≈0.83. To derive the error on
the fit parameters we estimate the total width of the global op-
timum for 10 Lac to be ∼0.152, i.e. the range of F=0.85...1.0
corresponds to the width of the optimum. In panel dof Fig. 14
we show the resulting distribution of Teffof the models within
this global optimum. In this figure we see that the maximum
variation, hence the error estimate, is ∼900 K, which is in
good agreement with the value derived using the fit diagram
of 10 Lac. For log gwe also find an error estimate of ∼0.1 dex,
which is also very similar to the value obtained with this di-
agram. The exact values as well as error estimates for all fit
parameters of all objects are given in Tab. 5. It is important
to note that our error analysis method also allows for an error
estimate of parameters to which the spectrum does not react
strongly. For 10 Lac this is clearly the case for the mass loss
rate, for which we find large error bars.
4.3. Derived parameters
In Tab. 5 the errors on the derived parameters were calculated
based on the error estimates of the fit parameters. Here we will
elaborate on their derivation.
The error in the stellar radii is dominated by the un-
certainty in the absolute visual magnitude. In case of the
Cyg OB2 objects we adopt these to be 0.1mconform the work
of Massey & Thompson (1991). For HD 15629, HD 217086
and ζOph we use the uncertainty as given by RPH of 0.3m.
The distance to 10 Lac and τSco was measured by Hipparcos.
Therefore, for these two objects we adopt the error based
on this measurement, which, respectively, is 0.4 and 0.2m.
Together with the uncertainty in Teffthe uncertainty in R⋆is
2In general this is not a fixed number. Considering all programme
stars we find the width of the global optimum to be within the range
∼0.1 to ∼0.2.
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 19
0.90
0.95
1.00
1.05
1.10
30.0 35.0 40.0 45.0
Teff ratio
Teff (kK)
Fig.15. Comparison of the effective temperatures obtained us-
ing automated fits (horizontal axis) and “by eye” fits. On the
vertical axis the ratio of automated relative to “by eye” tem-
perature determination is given. The dashed lines correspond
to a four percent error usually adopted for “by eye” determined
values.
calculated according to Eq. (8) of RPH, where we used the
largest absolute uncertainty in Tefffor a given object.
To correct the surface gravity for centrifugal forces, a cor-
rection conform Herrero et al. (1992) was applied to the gravity
determined from the spectral fits. This corrected value is given
in Tab. 4. As shown by RPH this correction has a non negligi-
ble effect on the error in the resulting log gc. Consequently, we
used their estimate to calculate the total error estimate of loggc
as given in Tab. 5. Using this error together with the uncertainty
in R⋆the resulting uncertainty in the spectroscopic mass was
calculated.
For the calculation of the uncertainty in the stellar luminos-
ity, we consistently adopted the largest absolute error in Teff.
The resulting ∆log L⋆as well as the uncertainty in Teffhave
an effect on the evolutionary mass. We have estimated errors
for this quantity using the error box spanned by ∆log L⋆and
∆log Teff.
5. Comparison with previous results
In this section we will compare the results obtained with our au-
tomated fitting method with those from “by eye” fits (relevant
references to the comparison studies are given in the previous
section). This does not constitute a one-to-one comparison of
the automated and “by eye” approach as this would require the
use of identical model atmosphere codes as well as the same set
of spectra, moreover, with identical continuum normalization.
Potential differences can therefore not exclusively be attributed
to the less bias sensitive automated fitting method. However,as
we have applied our method to a sizeable sample of early type
stars, the automated nature of it does assure that it is the most
homogeneous study to date, i.e. without at least some of the
biases involved in conventional analyses.
-0.4
-0.2
0.0
0.2
0.4
3.2 3.4 3.6 3.8 4.0 4.2
∆log g
log g
Fig. 16. Gravities obtained with automated fits (horizontal axis)
are compared to gravities determined from “by eye” fits. The
vertical axis gives the difference of the logarithm of the two
gravity determinations. Indicated by dashed lines are the 0.1
dex errors usually adopted for gravities determined “by eye”.
5.1. Effective temperature
In Fig. 15 a comparison of the effective temperatures deter-
mined in this study with Teffvalues obtained with “by eye” fits,
is presented. Indicated with dashed lines are the four percent
errors usually adopted for “by eye” fitted spectra. With the ex-
ception of the outlier HD 217086 at 38.1 kK, the agreement is
very good and no systematic trend is visible. From this plot we
can conclude that the Teffobtained with the automated fit is at
least as reliable as the temperatures determined in the conven-
tional way.
5.2. Gravities
In many cases the gravity obtained with the automated proce-
dure is significantly higher than the values obtained with the
conventional “by eye” fitted spectra. This is shown in Fig. 16,
where we show as a function of the gravities obtained in this
study the differences with the “by eye” determined values.
Indicated with dashed lines in this figure is the 0.1 dex error
in loggthat is often assigned to a “by eye” fitting of the hy-
drogen Balmer line wings. It is important to note that this plot
shows that there is no obvious trend in the differences, i.e. there
appears no systematic increase as a function of log gpresent.
It is clear, however, that there are three outliers for which
previous gravity determinations yield values that are at least
0.2 dex lower. These are in order of increasing gravity (as
determined in this study): Cyg OB2 #2, Cyg OB2 #7 and
HD 217086. For all three cases, previous spectroscopic mass
determinations result in values that are about a factor of two
less than the corresponding evolutionary masses. One reason
for these discrepant gravity values can be traced to a difference
between automated and “by eye” fitting. In “by eye” fitting, it
is custom to prohibit the theoretical line flux in the wings of
Balmer lines – specifically that at the position in the observed
line wing where the profile curvature is maximal – to be be-
20 M.R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
low that of the observed flux. This constraint has been used by
HPN for the Cyg OB2 stars; for HD 217086, we could not ver-
ify whether this was the case. The automated method does not
apply this constraint. Therefore, as it strives for a maximum fit-
ness, it tends to fit the curve through the signal noise as much
as possible. This yields a higher gravity.
A second reason is connected to the multidimensional na-
ture of the optimization problem. “By eye” fitting may not find
the optimum fit, as in general it can not simultaneously deal
in a sufficiently adequate way with all the free parameters of
the problem. Consequently, some of the “by eye” fitted spec-
tra do not correspond to the best fit possible. A good example
in which this appears to be the case is HD 217086. With the
automated fit we not only obtained a gravity that is higher by
∼0.3 dex, but also an effective temperature higher by 2.1 kK
compared to the results of RPH. Consequently, as the ioniza-
tion structure of the atmosphere depends heavily on this tem-
perature, so does the gravity one obtains from a spectral fit
for this temperature. As RPH obtained a gravity for a signif-
icantly lower effective temperature, the gravity obtained from
their spectral fit likely corresponds to the value from a local
optimum in parameter space.
5.3. Helium abundance and microturbulence
This analysis is the first in which the helium abundance and the
microturbulent velocity have been treated as continuous free
parameters. In the studies of HPN and RPH only two possible
values for the microturbulent velocity were adopted. For the he-
lium abundance an initial solar abundance was adopted, which
was modified when no satisfying fit could be obtained for this
abundance. Consequently, a comparison with these studies as
was done for e.g. the gravities, is not possible. Instead we will
only discuss whether the obtained values of these parameters
are reasonable and comment on possible correlations with other
parameters.
The helium abundances given in Tab. 4 show that no ex-
treme values were needed by the fitting method to obtain a
good fit. An exception to this may be YHe =0.21 obtained for
Cyg OB2 #7. However, as discussed earlier this value is still
significantly smaller than the YHe=0.3 obtained by HPN. With
respect to a possible relation between the helium abundance
and other parameters, only a small correlation between Teffand
YHe is found for the supergiants. For these objects it appears
(cf. Tab. 4) that the helium abundance increases with increas-
ing effective temperature. However, as we only analysed six
supergiants further investigation using a larger sample needs to
be undertaken.
Also for the microturbulent velocities no anomalous values
were needed to fit the spectra. The large error bars in the turbu-
lent velocity quoted in Tab. 5, especially for the supergiants,
show that the profiles are not very sensitive to this parame-
ter. This is consistent with the study of Villamariz & Herrero
(2000) and RPH. The entries given on some of the positive
errors in the table indicate that they reach up to the maximum
allowed value of vturb, which is 20 km s−1. Therefore, they are
formally not defined. The fact that some of the small scale tur-
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
-7.2 -6.8 -6.4 -6.0 -5.6 -5.2 -4.8
∆ log dM/dt
log dM/dt
Fig.17. Difference between mass loss rate obtained by the au-
tomated method (given by the horizontal) axis and values de-
termined by eye. A typical 0.15 dex error is indicated by the
dashed lines. The two outliers at log ˙
M≃ −7.2 and and log ˙
M≃
−6.8, respectively, correspond to 10 Lac and Cyg OB2 #2.
bulent velocities are close to this maximum value may indicate
that they represent lower limits, though, again, this likely re-
flects that they are poorly constraint.
No correlation of the microturbulence with any of the other
parameters is found. In particular not between vturb and log g
and vturb and YHe. Various authors have hinted at such a corre-
lation (e.g. Kilian 1992).
5.4. Wind parameters
The straightforward comparison of the mass loss rates obtained
with the automated method with values determined from spec-
tral fits “by eye” is shown in Fig. 17. With exception of τSco
at log ˙
M=−7.2, for which the mass loss rate determined by
Gathier et al. (1981) from UV line fitting serves as a compar-
ison, all mass loss rates are compared to values determined
from Hαfitting. For this comparison we assume an error of
0.15 dex in the “by eye” determined values. This uncertainty
corresponds to a typical error obtained from Hαfitting and is
shown in Fig. 17 as a set of dashed lines. With exception of
10 Lac and τSco, for which the mass loss rate determination
is uncertain, this error is also comparable to the errors obtained
with the automated method.
Two objects show a relative increase in ˙
Mwhich is much
larger than the typical error. These are 10 Lac at log ˙
M≃ −7.3
and Cyg OB2 #2 at log ˙
M≃ −6.8. In the case of the latter we
showed that the increase is due to a more efficient use of wind
information stored in the line profiles by the automated method,
which improves the relation of Cyg OB2 #2 with respect to the
wind-momentum relation (see Sect. 6.2).
With respect to 10 Lac we already mentioned that a range
of more than two orders of magnitude in mass loss rate has
been found in different studies. Here we have made the com-
parison with the upper limit found by HPN, which corresponds
to one of the lowest ˙
Mdetermined for this object. If we would
have compared our findings to the higher value obtained by
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 21
0
20
40
60
80
100
120
0 20 40 60 80 100 120
Evolutionary Mass (Msun)
Spectroscopic Mass (Msun)
Fig.18. Spectroscopic masses derived in this study compared
to evolutionary masses from Schaller et al. (1992). With the
gravities obtained from the automated fits no mass discrepancy
is found and no systematic deviation between the spectroscopi-
cally derived masses and the evolutionary predicted masses can
be observed.
Howarth & Prinja (1989), 10 Lac would be at ∆˙
M=−0.5, i.e.
the situation in Fig. 17 would be reversed. Consequently, the
large difference for 10 Lac shown in this figure can not be as-
signed to an error in the automated method, but rather reflects
our limited understanding of this object.
All in all, we can conclude that the general agreement be-
tween mass loss rates obtained with the automated method and
“by eye” determinations is very good.
6. Implications for the properties of massive stars
With our automated method we have analysed a sizeable sam-
ple of early type stars in a homogeneous way, which allows
a first discussion of the implications the newly obtained pa-
rameters may have on the mass and modified wind-momentum
luminosity relation (WLR) of massive stars. A thorough dis-
cussion however needs to be based on a much larger sample,
therefore at this point we keep the discussion general and the
conclusions tentative.
6.1. On the mass discrepancy
The so called mass discrepancy was first noticed by
Herrero et al. (1992). These authors found that the spectro-
scopic masses, i.e. masses calculated from the spectroscopi-
cally determined gravity, were systematically smaller than the
masses predicted by evolutionary calculations. The situation
improved considerably with the use of unified stellar atmo-
sphere models (e.g. Herrero et al. 2002). However, as pointed
out by Repolust et al. (2004) for stars with masses lower than
50 M⊙still a milder form of a mass discrepancy appears to
persist.
Does the automated fitting method, employing the latest
version of , help in resolving the mass discrepancy?
In Fig. 18 we present a comparison of the spectroscopic masses
25
26
27
28
29
30
4 4.5 5 5.5 6 6.5
log MWM
log(L/L )
Fig.19. Modified wind momentum (MWM) in units of
[g cms−2R⊙] of the objects fitted with the automated method
(solid dots). The solid line, giving the wind-momentum lumi-
nosity relation (WLR), corresponds to the regression of the
modified wind momenta. Given by the dashed line is the pre-
dicted WLR of Vink et al. (2000).
calculated with the gravities obtained in this study, with masses
derived by interpolating evolutionary tracks of Schaller et al.
(1992). It is clear that with the new gravities the situation is
very satisfying. All objects have spectroscopic and evolution-
ary masses which agree within the error bars.
For stars with masses below 50 M⊙a milder form of the
mass discrepancy (as found by RPH; see their Fig. 20) could
still be present, but with the present data no systematic offset
between the two mass scales can be appreciated. Though we
feel it may be premature to conclude that the present analysis
shows that the mass discrepancy has been resolved, our results
point to a clear improvement.
6.2. Wind-momentum luminosity relation
The modified stellar wind momentum (MWM) versus luminos-
ity relation offers a meaningful way to compare observed wind
properties with aspects and predictions of the theory of line
driven winds (see Kudritzki & Puls 2000 for a comprehensive
discussion). Without going into any detail, the modified wind
momentum Dmom =˙
Mv∞R1/2
⋆is predicted to be a power law of
stellar luminosity.
log Dmom =xlog(L⋆/L⊙)+log D◦,(7)
where x, the inverse of the slope of the line-strength distribu-
tion function corrected for ionization effects (Puls et al. 2000),
is expected to be a function of spectral type and metal abun-
dance, and D◦is a function of metallicity and possibly lumi-
nosity class (Markova et al. 2004). The advantageous property
of Dmom is that it is not very sensitive to the stellar mass.
The limited number of stars studied in this paper is clearly
insufficient to disentangle subtleties in the Dmom vs. L⋆rela-
tion. However, it is interesting to compare the observed and
predicted modified wind momentum, as well as to discuss the
location of 10 Lac – a notorious outlier.
22 M.R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
Figure 19 shows this comparison between derived and the-
oretical modified wind momentum. Using all programme stars
to construct an empirical linear curve in the units of this dia-
gram gives the following relation
log Dmom =(1.88 ±0.09) log(L⋆/L⊙)+(18.59 ±0.52) .(8)
Within the given errors this relation is equal to the theoreti-
cal WLR predicted by Vink et al. (2000), who found x=1.83
and log Dmom =18.68. Note that the low luminosity objects
(log L⋆/L⊙.5.5) also follow the average relation. Therefore,
the newly obtained mass loss rates do not show the discrepancy
found by Puls et al. (1996) and Kudritzki & Puls (2000), but
confirm the work of RPH. These authors found the low lumi-
nosity objects to follow the general trend, based on upper limits
they obtained for the mass loss rates, whereas our new method
is sensitive enough to determine these self-consistently.
In order to investigate the effect of the anomalously low
Dmom obtained for 10 Lac, we also constructed a WLR exclud-
ing this object. We found that for this new relation the parame-
ters xand log D◦only changed with ∼0.02 and ∼0.01, respec-
tively, reflecting the large error bars found for this object.
Previous investigations by Markova et al. (2004) and RPH
have found the WLR to be as function of luminosity class.
Whereas the former study finds a steeper WLR for the su-
pergiants compared to the dwarfs, the latter finds the opposite
(though RPH remark that the subset of Cyg OB2 stars seem
to behave more in accordance with the theoretical result). In
our sample no obvious separation is visible. In particular note
the two objects overlapping at log L⋆/L⊙=4.9 in Fig. 19,
which are the dwarf ζOph and the supergiant Cyg OB2 #2.
To investigate a possible separation in more detail, a separate
WLR was constructed for the Cyg OB2 supergiants. The re-
sulting values of the parameters obtained are x=1.79 ±0.14
and log D◦=19.12 ±0.80. The decrease in xqualitatively con-
firms the work of Markova et al.. However, we have to realize
that our sample might be too small from a statistical point of
view to be able to firmly conclude whether a real separation
exists. Therefore, this question has to be postponed until we
have analysed a larger sample.
7. Summary, conclusions and future work
We have presented the first method for the automated fitting of
spectra of massive stars with stellar winds. In this first imple-
mentation, a set of continuum normalized optical spectral lines
is fitted to predictions made with the fast performance non-LTE
model atmosphere code by Puls et al. (2005). The fit-
ting method itself is based on the genetic algorithm by
Charbonneau (1995), which was parallelized in order to handle
the thousands of models which have to be calculated
for an automated fit. Concerning the automated method we can
draw the following conclusions:
i) The method is robust. In applying the method to a num-
ber of formal tests, to the study of seven O-type stars in
Cyg OB2, and to five Galactic stars including extreme rota-
tors and/or stars with weak winds (few times 10−8M⊙yr−1)
the fitting procedure did not encounter convergence prob-
lems.
ii) Using the width of the global optimum in terms of fitness,
defining the group of best fitting models, we are able to de-
fine error estimates for all of the six free parameters of the
model (Teff, log g, helium over hydrogen abundance, vturb,
˙
Mand β). These errors compare well with errors adopted
in “by eye” fitting methods.
iii) For the investigated dataset our automated fitting method
recovers mass-loss rates down to ∼6×10−8M⊙yr−1to
within an error of a factor of two. We point out that even
for such low mass-loss rates it is not only the core of the
hydrogen Hαline that is a mass-loss diagnostics. When ig-
noring this core the GA still recovers ˙
M, showing that the
GA is also sensitive to indirect effects of a change in ˙
M
on the atmospheric structure as a whole. However, for the
method to fully take advantage of this information a very
accurate continuum normalization is required.
iv) Though we have so far tested our method for O-type stars
and early B-type dwarf stars, the method can also be ap-
plied to B and A supergiants when atomic models of diag-
nostic lines (such as Si and Si ) are implemented into
the analysis.
We have re-investigated seven O-type stars in the young clus-
ter Cyg OB2 and compared our results with the study by
Herrero, Puls, & Najarro (2002). The HPN study uses an ear-
lier version of and a “by eye” fitting procedure. The
only difference between the two studies in terms of the treat-
ment of the free parameters is that HPN did not treat the mi-
croturbulent velocity and the hydrogen over helium abundance
ratio as continuous free parameters. Instead, they opted to adopt
in case of the former two possible values only. In case of the
latter an initial solar abundance was adopted which was mod-
ified in case no satisfying fit could be obtained with this solar
abundance. We have also compared the results of an automated
fitting to five early-type dwarf stars to further investigate the
robustness of our method for stars with high rotational veloci-
ties and/or low mass loss rates. With respect to weak winds we
refer to conclusion iii. Regarding large vrsin ivalues, we find
that these do not pose problems for the automated method. This
is reflected in conclusion i. Concerning the spectral analysis of
the entire sample we can draw the following conclusions:
v) For almost all parameters we find excellent agreement with
the results of HPN and RPH, which, we note, make use
of a previous version of and an independent con-
tinuum normalization. The quality of our fits (in terms of
fitness, which is a measure for the χ2of the lines) is even
better than obtained in these prior studies.
vi) In three cases we find a significantly higher surface gravity
(by up to 0.36 dex). We identify two possible causes for this
difference that may be connected to the difference between
automated and “by eye” fitting. First, comparison of the two
methods indicates that in fitting the Balmer line wings the
latter method places essentially infinite strength to the ob-
served flux at the point of maximum curvature of the wing
profile. The automated method does not do this. Second, as
the automated method is a multidimensional optimization
method it may simply find a better fit to the overall spec-
M. R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method 23
trum. In at least one case this implied a higher temperature
and significantly higher gravity.
vii) A comparison of our derived masses with those predicted
by evolutionary calculations does not show any system-
atic discrepancy. Such a discrepancy was first noted by
Herrero et al. (1992), though was partly resolved when
model atmospheres improved (e.g. see HPN). Still, with
state-of-the-art models a mild form of a mass discrepancy
remained for stars with masses below 50 M⊙(e.g. see
RPH). The automated fitting approach in combination with
the improved version of does not find evidence for
a mass discrepancy, although we remark that a truly robust
conclusion, particularly for stars between 20 and 50 solar
masses, may require the investigation of a larger sample.
viii) The empirical modified wind momentum relation con-
structed on the basis of the twelve objects analysed in this
study agree to within the error bars with the theoretical
MWM relations based on the Vink et al. (2000) predictions
of mass loss rates.
This first implementation of a genetic algorithm combined
with the fast performance code already shows the
high potential of automatic spectral analysis. With the current
rapid increase in observations of early-type massive stars the
need for an automated fitting method is evident. We will first
use our method to analyse the ∼100 O-type and early B-type
stars observed in the VLT large programme FLAMES Survey
of Massive Stars (Evans et al. 2005) in the Galaxy and the
Magellanic Clouds in a homogeneous way. Future develop-
ment of the automated fitting method is likely to be in conjunc-
tion with the further development of . Improvements
will include the modeling of: near-infrared lines (see e.g.
Lenorzer et al. 2004 and Repolust et al. 2005), optical CNO
lines (see e.g. Trundle et al. 2004), and possibly UV resonance
lines. Additional model parameters that may be constrained
within an automated approach include a depth dependent pro-
file for the microturbulent velocity and small scale clumping.
Within the current implementation, most likely the method will
also be able to constrain the terminal flow velocity of A-type
supergiants (McCarthy et al. 1997; Kudritzki et al. 1999).
Acknowledgements. We would like to thank Chris Evans, Ian Hunter,
Stephen Smartt and Wing-Fai Thi for constructive discussions, and
Michiel Min for sharing his insights in automated fitting. M.R.M. ac-
knowledges financial support from the NWO Council for Physical
Sciences. F.N. acknowledges PNAYA2003-02785-E and AYA2004-
08271-C02-02 grants and the Ramon y Cajal program.
References
Becker, S. R. & Butler, K. 1992, A&A, 265, 647
Bianchi, L. & Garcia, M. 2002, ApJ, 581, 610
Bouret, J.-C., Lanz, T., Hillier, D. J., et al. 2003, ApJ, 595, 1182
Charbonneau, P. 1995, ApJS, 101, 309
Charbonneau, P. & Knapp, B. 1995, NCAR Technical Note 418+IA
Conti, P. S. & Alschuler, W. R. 1971, ApJ, 170, 325
Crowther, P. A., Hillier, D. J., Evans, C. J., et al. 2002, ApJ, 579, 774
Darwin, C. 1859, On the Origin of Species by Means of Natural
Selection (John Murray, London)
De Becker, M., Rauw, G., & Manfroid, J. 2004, A&A, 424, L39
de Jager, C., Nieuwenhuijzen, H., & van der Hucht, K. A. 1988,
A&AS, 72, 259
de Koter, A., Heap, S. R., & Hubeny, I. 1997, ApJ, 477, 792
Evans, C. J., Crowther, P. A., Fullerton, A. W., & Hillier, D. J. 2004,
ApJ, 610, 1021
Evans, C. J., Smart, S. J., Lee, J. K., et al. 2005, A&A, 437, 467
Garcia, M. & Bianchi, L. 2004, ApJ, 606, 497
Gathier, R., Lamers, H. J. G. L. M., & Snow, T. P. 1981, ApJ, 247,
173
Gibson, S. E. & Charbonneau, P. 1998, J. Geophys. Res., 103, 14511
Grevesse, N. & Sauval, A. J. 1998, Space Science Reviews, 85, 161
Groenewegen, M. A. T. & Lamers, H. J. G. L. M. 1989, A&AS, 79,
359
Hamann, W.-R. 1981, A&A, 100, 169
Herrero, A. 1993, Space Science Reviews, 66, 137
Herrero, A., Corral, L. J., Villamariz, M. R., & Mart´ın, E. L. 1999,
A&A, 348, 542
Herrero, A., Kudritzki, R. P., Vilchez, J. M., et al. 1992, A&A, 261,
209
Herrero, A., Puls, J., Corral, L. J., Kudritzki, R. P., & Villamariz, M. R.
2001, A&A, 366, 623
Herrero, A., Puls, J., & Najarro, F. 2002, A&A, 396, 949
Herrero, A., Puls, J., & Villamariz, M. R. 2000, A&A, 354, 193
Hillier, D. J. 1991, A&A, 247, 455
Hillier, D. J., Lanz, T., Heap, S. R., et al. 2003, ApJ, 588, 1039
Hillier, D. J. & Miller, D. L. 1998, ApJ, 496, 407
Hillier, D. J. & Miller, D. L. 1999, ApJ, 519, 354
Howarth, I. D. & Prinja, R. K. 1989, ApJS, 69, 527
Howarth, I. D., Siebert, K. W., Hussain, G. A. J., & Prinja, R. K. 1997,
MNRAS, 284, 265
Hubeny, I. & Lanz, T. 1995, ApJ, 439, 875
Humphreys, R. M. 1978, ApJS, 38, 309
Kilian, J. 1992, A&A, 262, 171
Kilian, J., Becker, S. R., Gehren, T., & Nissen, P. E. 1991, A&A, 244,
419
Kudritzki, R. & Puls, J. 2000, ARA&A, 38, 613
Kudritzki, R.-P. 1980, A&A, 85, 174
Kudritzki, R. P., Puls, J., Lennon, D. J., et al. 1999, A&A, 350, 970
Kudritzki, R. P. & Simon, K. P. 1978, A&A, 70, 653
Lenorzer, A., Mokiem, M. R., de Koter, A., & Puls, J. 2004, A&A,
422, 275
Markova, N., Puls, J., Repolust, T., & Markov, H. 2004, A&A, 413,
693
Martins, F., Schaerer, D., Hillier, D. J., & Heydari-Malayeri, M. 2004,
A&A, 420, 1087
Massey, P., Bresolin, F., Kudritzki, R. P., Puls, J., & Pauldrach,
A. W. A. 2004, ApJ, 608, 1001
Massey, P. & Thompson, A. B. 1991, AJ, 101, 1408
Matthews, T. A. & Sandage, A. R. 1963, ApJ, 138, 30
McCarthy, J. K., Kudritzki, R., Lennon, D. J., Venn, K. A., & Puls, J.
1997, ApJ, 482, 757
Metcalfe, T. S. & Charbonneau, P. 2003, Journal of Computational
Physics, 185, 176
Metcalfe, T. S.,Nather, R. E., & Winget, D. E. 2000, ApJ, 545, 974
Mokiem, M. R., Mart´ın-Hern´andez, N. L., Lenorzer, A., de Koter, A.,
& Tielens, A. G. G. M. 2004, A&A, 419, 319
Pauldrach, A., Puls, J., & Kudritzki, R. P. 1986, A&A, 164, 86
Pauldrach, A. W. A., Hoffmann, T. L., & Lennon, M. 2001, A&A, 375,
161
Puls, J., Kudritzki, R.-P., Herrero, A., et al. 1996, A&A, 305, 171
Puls, J., Springmann, U., & Lennon, M. 2000, A&AS, 141, 23
Puls, J., Urbaneja, M. A., Venero, R., et al. 2005, A&A, 435, 669
Repolust, T., Puls, J., Hanson, M. M., Kudritzki, R.-P., & Mokiem,
24 M.R. Mokiem et al.: Spectral analysis using a genetic algorithm based fitting method
M. R. 2005, A&A, accepted
Repolust, T., Puls, J., & Herrero, A. 2004, A&A, 415, 349
Schaller, G., Schaerer, D., Meynet, G., & Maeder, A. 1992, A&AS,
96, 269
Sch¨onberner, D., Herrero, A., Becker, S., et al. 1988, A&A, 197, 209
Schr¨oder, S. E., Kaper, L., Lamers, H. J. G. L. M., & Brown, A. G. A.
2004, A&A, 428, 149
Smith, K. C. & Howarth, I. D. 1998, MNRAS, 299, 1146
Trundle, C., Lennon, D. J., Puls, J., & Dufton, P. L. 2004, A&A, 417,
217
Villamariz, M. R. & Herrero, A. 2000, A&A, 357, 597
Villamariz, M. R. & Herrero, A. 2005, A&A, accepted
Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2000, A&A, 362,
295
Voels, S. A., Bohannan, B., Abbott, D. C., & Hummer, D. G. 1989,
ApJ, 340, 1073
Walborn, N. R. 1972, AJ, 77, 312
Walborn, N. R. 1973, AJ, 78, 1067
Zaal, P. A., de Koter, A., Waters, L. B. F. M., et al. 1999, A&A, 349,
573
